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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembnj1090 30301* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   (𝜂′[𝑗 / 𝑖]𝜂)    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))    &   ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))       ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 (𝜌𝜂))

Theorembnj1093 30302* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑗(((𝜃𝜏𝜒) ∧ 𝜑0) → (𝑓𝑖) ⊆ 𝐵)    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))       ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑗(𝜑0 → (𝑓𝑖) ⊆ 𝐵))

Theorembnj1097 30303 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝑖 = ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)

Theorembnj1110 30304* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   𝐷 = (ω ∖ {∅})    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))    &   (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))       𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑗𝑛𝑖 = suc 𝑗 ∧ (𝑓𝑗) ⊆ 𝐵))

Theorembnj1112 30305* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))       (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))

Theorembnj1118 30306* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   𝐷 = (ω ∖ {∅})    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))    &   (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))       𝑗((𝑖 ≠ ∅ ∧ ((𝜃𝜏𝜒) ∧ 𝜑0)) → (𝑓𝑖) ⊆ 𝐵)

Theorembnj1121 30307 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   ((𝜃𝜏𝜒𝜁) → ∀𝑖𝑛 𝜂)    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}       ((𝜃𝜏𝜒𝜁) → 𝑧𝐵)

Theorembnj1123 30308* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   (𝜂′[𝑗 / 𝑖]𝜂)       (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))

Theorembnj1030 30309* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))    &   (𝜁 ↔ (𝑖𝑛𝑧 ∈ (𝑓𝑖)))    &   𝐷 = (ω ∖ {∅})    &   𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))    &   (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))    &   (𝜑′[𝑗 / 𝑖]𝜑)    &   (𝜓′[𝑗 / 𝑖]𝜓)    &   (𝜒′[𝑗 / 𝑖]𝜒)    &   (𝜃′[𝑗 / 𝑖]𝜃)    &   (𝜏′[𝑗 / 𝑖]𝜏)    &   (𝜁′[𝑗 / 𝑖]𝜁)    &   (𝜂′[𝑗 / 𝑖]𝜂)    &   (𝜎 ↔ ((𝑗𝑛𝑗 E 𝑖) → 𝜂′))    &   (𝜑0 ↔ (𝑖𝑛𝜎𝑓𝐾𝑖 ∈ dom 𝑓))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Theorembnj1124 30310 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝜃𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Theorembnj1133 30311* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))    &   ((𝑖𝑛𝜏) → 𝜃)       (𝜒 → ∀𝑖𝑛 𝜃)

Theorembnj1128 30312* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ (𝜒 → (𝑓𝑖) ⊆ 𝐴))    &   (𝜏 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜃))    &   (𝜑′[𝑗 / 𝑖]𝜑)    &   (𝜓′[𝑗 / 𝑖]𝜓)    &   (𝜒′[𝑗 / 𝑖]𝜒)    &   (𝜃′[𝑗 / 𝑖]𝜃)       (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)

Theorembnj1127 30313 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌𝐴)

Theorembnj1125 30314 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴𝑌 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑌, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))

Theorembnj1145 30315* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}    &   (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))    &   (𝜃 ↔ ((𝑖 ≠ ∅ ∧ 𝑖𝑛𝜒) ∧ (𝑗𝑛𝑖 = suc 𝑗)))        trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Theorembnj1147 30316 Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴

Theorembnj1137 30317* Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo(𝐵, 𝐴, 𝑅))

Theorembnj1148 30318 Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝑋𝐴) → pred(𝑋, 𝐴, 𝑅) ∈ V)

Theorembnj1136 30319* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)

Theorembnj1152 30320 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))

Theorembnj1154 30321* Property of Fr. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 Fr 𝐴𝐵𝐴𝐵 ≠ ∅ ∧ 𝐵 ∈ V) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

Theorembnj1171 30322 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓) → 𝐵𝐴)    &   𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))       𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))

Theorembnj1172 30323 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))    &   ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))       𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐴 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))

Theorembnj1173 30324 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))    &   ((𝜑𝜓) → 𝑅 FrSe 𝐴)    &   ((𝜑𝜓) → 𝑋𝐴)       ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))

Theorembnj1174 30325 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))    &   (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))       𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))

Theorembnj1175 30326 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   (𝜒 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ (𝑤𝐴𝑤𝑅𝑧)))    &   (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))       (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))

Theorembnj1176 30327* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))    &   ((𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V) → ∃𝑧𝐶𝑤𝐶 ¬ 𝑤𝑅𝑧)       𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))

Theorembnj1177 30328 Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ (𝑋𝐵𝑦𝐵𝑦𝑅𝑋))    &   𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)    &   ((𝜑𝜓) → 𝑅 FrSe 𝐴)    &   ((𝜑𝜓) → 𝐵𝐴)    &   ((𝜑𝜓) → 𝑋𝐴)       ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))

Theorembnj1186 30329* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑧𝑤((𝜑𝜓) → (𝑧𝐵 ∧ (𝑤𝐵 → ¬ 𝑤𝑅𝑧)))       ((𝜑𝜓) → ∃𝑧𝐵𝑤𝐵 ¬ 𝑤𝑅𝑧)

Theorembnj1190 30330* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅))    &   (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))       ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)

Theorembnj1189 30331* Technical lemma for bnj69 30332. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ (𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅))    &   (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))    &   (𝜒 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)       (𝜑 → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

21.4.3  The existence of a minimal element in certain classes

Theorembnj69 30332* Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

Theorembnj1228 30333* Existence of a minimal element in certain classes: if 𝑅 is well-founded and set-like on 𝐴, then every nonempty subclass of 𝐴 has a minimal element. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑤𝐵 → ∀𝑥 𝑤𝐵)       ((𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)

21.4.4  Well-founded induction

Theorembnj1204 30334* Well-founded induction. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))       ((𝑅 FrSe 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)

Theorembnj1234 30335* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐷 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}       𝐶 = 𝐷

Theorembnj1245 30336* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))    &   𝑍 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐾 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑍))}       (𝜑 → dom 𝐴)

Theorembnj1256 30337* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜑 → ∃𝑑𝐵 𝑔 Fn 𝑑)

Theorembnj1259 30338* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜑 → ∃𝑑𝐵 Fn 𝑑)

Theorembnj1253 30339* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜑𝐸 ≠ ∅)

Theorembnj1279 30340* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       ((𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥) → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)

Theorembnj1286 30341* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))       (𝜓 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝐷)

Theorembnj1280 30342* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))    &   (𝜓 → ( pred(𝑥, 𝐴, 𝑅) ∩ 𝐸) = ∅)       (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))

Theorembnj1296 30343* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )    &   𝐸 = {𝑥𝐷 ∣ (𝑔𝑥) ≠ (𝑥)}    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑔𝐶𝐶 ∧ (𝑔𝐷) ≠ (𝐷)))    &   (𝜓 ↔ (𝜑𝑥𝐸 ∧ ∀𝑦𝐸 ¬ 𝑦𝑅𝑥))    &   (𝜓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = ( ↾ pred(𝑥, 𝐴, 𝑅)))    &   𝑍 = ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐾 = {𝑔 ∣ ∃𝑑𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑔𝑥) = (𝐺𝑍))}    &   𝑊 = ⟨𝑥, ( ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐿 = { ∣ ∃𝑑𝐵 ( Fn 𝑑 ∧ ∀𝑥𝑑 (𝑥) = (𝐺𝑊))}       (𝜓 → (𝑔𝑥) = (𝑥))

Theorembnj1309 30344* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}       (𝑤𝐵 → ∀𝑥 𝑤𝐵)

Theorembnj1307 30345* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝑤𝐵 → ∀𝑥 𝑤𝐵)       (𝑤𝐶 → ∀𝑥 𝑤𝐶)

Theorembnj1311 30346* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )       ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))

Theorembnj1318 30347 Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑋 = 𝑌 → trCl(𝑋, 𝐴, 𝑅) = trCl(𝑌, 𝐴, 𝑅))

Theorembnj1326 30348* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐷 = (dom 𝑔 ∩ dom )       ((𝑅 FrSe 𝐴𝑔𝐶𝐶) → (𝑔𝐷) = (𝐷))

Theorembnj1321 30349* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))       ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏) → ∃!𝑓𝜏)

Theorembnj1364 30350 Property of FrSe. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑅 FrSe 𝐴𝑅 Se 𝐴)

Theorembnj1371 30351* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))       (𝑓𝐻 → Fun 𝑓)

Theorembnj1373 30352* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   (𝜏′[𝑦 / 𝑥]𝜏)       (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))

Theorembnj1374 30353* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}       (𝑓𝐻𝑓𝐶)

Theorembnj1384 30354* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻       (𝑅 FrSe 𝐴 → Fun 𝑃)

Theorembnj1388 30355* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)       (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)

Theorembnj1398 30356* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   (𝜃 ↔ (𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   (𝜂 ↔ (𝜃𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))       (𝜒 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)

Theorembnj1413 30357* Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝑋𝐴) → 𝐵 ∈ V)

Theorembnj1408 30358* Technical lemma for bnj1414 30359. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   𝐶 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))    &   (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴))    &   (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))       ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)

Theorembnj1414 30359* Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = ( pred(𝑋, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑋, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))       ((𝑅 FrSe 𝐴𝑋𝐴) → trCl(𝑋, 𝐴, 𝑅) = 𝐵)

Theorembnj1415 30360* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻       (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))

Theorembnj1416 30361 Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))       (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Theorembnj1418 30362 Property of pred. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)

Theorembnj1417 30363* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
(𝜑𝑅 FrSe 𝐴)    &   (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜓))    &   (𝜃 ↔ (𝜑𝑥𝐴𝜒))    &   𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))       (𝜑 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Theorembnj1421 30364* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   (𝜒 → Fun 𝑃)    &   (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))       (𝜒 → Fun 𝑄)

Theorembnj1444 30365* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))       (𝜌 → ∀𝑦𝜌)

Theorembnj1445 30366* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))    &   (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))    &   𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩       (𝜎 → ∀𝑑𝜎)

Theorembnj1446 30367* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩       ((𝑄𝑧) = (𝐺𝑊) → ∀𝑑(𝑄𝑧) = (𝐺𝑊))

Theorembnj1447 30368* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩       ((𝑄𝑧) = (𝐺𝑊) → ∀𝑦(𝑄𝑧) = (𝐺𝑊))

Theorembnj1448 30369* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩       ((𝑄𝑧) = (𝐺𝑊) → ∀𝑓(𝑄𝑧) = (𝐺𝑊))

Theorembnj1449 30370* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))       (𝜁 → ∀𝑓𝜁)

Theorembnj1442 30371* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))       (𝜂 → (𝑄𝑧) = (𝐺𝑊))

Theorembnj1450 30372* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜃 ↔ (𝜒𝑧𝐸))    &   (𝜂 ↔ (𝜃𝑧 ∈ {𝑥}))    &   (𝜁 ↔ (𝜃𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))    &   (𝜌 ↔ (𝜁𝑓𝐻𝑧 ∈ dom 𝑓))    &   (𝜎 ↔ (𝜌𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))    &   (𝜑 ↔ (𝜎𝑑𝐵𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌)))    &   𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩       (𝜁 → (𝑄𝑧) = (𝐺𝑊))

Theorembnj1423 30373* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑃 Fn trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))       (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))

Theorembnj1452 30374* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))       (𝜒𝐸𝐵)

Theorembnj1466 30375* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})       (𝑤𝑄 → ∀𝑓 𝑤𝑄)

Theorembnj1467 30376* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})       (𝑤𝑄 → ∀𝑑 𝑤𝑄)

Theorembnj1463 30377* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))    &   (𝜒𝑄 ∈ V)    &   (𝜒 → ∀𝑧𝐸 (𝑄𝑧) = (𝐺𝑊))    &   (𝜒𝑄 Fn 𝐸)    &   (𝜒𝐸𝐵)       (𝜒𝑄𝐶)

Theorembnj1489 30378* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})       (𝜒𝑄 ∈ V)

Theorembnj1491 30379* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   (𝜒 → (𝑄𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))       ((𝜒𝑄 ∈ V) → ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))

Theorembnj1312 30380* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))    &   𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}    &   (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))    &   (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))    &   (𝜏′[𝑦 / 𝑥]𝜏)    &   𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}    &   𝑃 = 𝐻    &   𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})    &   𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩    &   𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))       (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Theorembnj1493 30381* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       (𝑅 FrSe 𝐴 → ∀𝑥𝐴𝑓𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Theorembnj1497 30382* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       𝑔𝐶 Fun 𝑔

Theorembnj1498 30383* Technical lemma for bnj60 30384. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

21.4.5  Well-founded recursion, part 1 of 3

Theorembnj60 30384* Well-founded recursion, part 1 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       (𝑅 FrSe 𝐴𝐹 Fn 𝐴)

Theorembnj1514 30385* Technical lemma for bnj1500 30390. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}       (𝑓𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓𝑥) = (𝐺𝑌))

Theorembnj1518 30386* Technical lemma for bnj1500 30390. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))    &   (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))       (𝜓 → ∀𝑑𝜓)

Theorembnj1519 30387* Technical lemma for bnj1500 30390. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Theorembnj1520 30388* Technical lemma for bnj1500 30390. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       ((𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Theorembnj1501 30389* Technical lemma for bnj1500 30390. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝑥𝐴))    &   (𝜓 ↔ (𝜑𝑓𝐶𝑥 ∈ dom 𝑓))    &   (𝜒 ↔ (𝜓𝑑𝐵 ∧ dom 𝑓 = 𝑑))       (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

21.4.6  Well-founded recursion, part 2 of 3

Theorembnj1500 30390* Well-founded recursion, part 2 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       (𝑅 FrSe 𝐴 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Theorembnj1525 30391* Technical lemma for bnj1522 30394. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   (𝜓 ↔ (𝜑𝐹𝐻))       (𝜓 → ∀𝑥𝜓)

Theorembnj1529 30392* Technical lemma for bnj1522 30394. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜒 → ∀𝑥𝐴 (𝐹𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))    &   (𝑤𝐹 → ∀𝑥 𝑤𝐹)       (𝜒 → ∀𝑦𝐴 (𝐹𝑦) = (𝐺‘⟨𝑦, (𝐹 ↾ pred(𝑦, 𝐴, 𝑅))⟩))

Theorembnj1523 30393* Technical lemma for bnj1522 30394. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶    &   (𝜑 ↔ (𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))    &   (𝜓 ↔ (𝜑𝐹𝐻))    &   (𝜒 ↔ (𝜓𝑥𝐴 ∧ (𝐹𝑥) ≠ (𝐻𝑥)))    &   𝐷 = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐻𝑥)}    &   (𝜃 ↔ (𝜒𝑦𝐷 ∧ ∀𝑧𝐷 ¬ 𝑧𝑅𝑦))       ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)

21.4.7  Well-founded recursion, part 3 of 3

Theorembnj1522 30394* Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}    &   𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩    &   𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}    &   𝐹 = 𝐶       ((𝑅 FrSe 𝐴𝐻 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐻𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → 𝐹 = 𝐻)

21.5  Mathbox for Mario Carneiro

21.5.1  Predicate calculus with all distinct variables

Axiomax-7d 30395* Distinct variable version of ax-11 2021. (Contributed by Mario Carneiro, 14-Aug-2015.)
(∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Axiomax-8d 30396* Distinct variable version of ax-7 1922. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Axiomax-9d1 30397 Distinct variable version of ax-6 1875, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑥

Axiomax-9d2 30398* Distinct variable version of ax-6 1875, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
¬ ∀𝑥 ¬ 𝑥 = 𝑦

Axiomax-10d 30399* Distinct variable version of axc11n 2295. (Contributed by Mario Carneiro, 14-Aug-2015.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Axiomax-11d 30400* Distinct variable version of ax-12 2034. (Contributed by Mario Carneiro, 14-Aug-2015.)
(𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

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