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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oieq2 8301 | Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ (𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵)) | ||
Theorem | nfoi 8302 | Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴) | ||
Theorem | ordiso2 8303 | Generalize ordiso 8304 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵) | ||
Theorem | ordiso 8304* | Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵))) | ||
Theorem | ordtypecbv 8305* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) ⇒ ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹 | ||
Theorem | ordtypelem1 8306* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) | ||
Theorem | ordtypelem2 8307* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → Ord 𝑇) | ||
Theorem | ordtypelem3 8308* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣}) | ||
Theorem | ordtypelem4 8309* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) | ||
Theorem | ordtypelem5 8310* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) | ||
Theorem | ordtypelem6 8311* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ 𝑀 → (𝑂‘𝑁)𝑅(𝑂‘𝑀))) | ||
Theorem | ordtypelem7 8312* | Lemma for ordtype 8320. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝑂)) | ||
Theorem | ordtypelem8 8313* | Lemma for ordtype 8320. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) | ||
Theorem | ordtypelem9 8314* | Lemma for ordtype 8320. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8318 implies that either ran 𝑂 ⊆ 𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑂 ∈ V) ⇒ ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) | ||
Theorem | ordtypelem10 8315* | Lemma for ordtype 8320. Using ax-rep 4699, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) | ||
Theorem | oi0 8316 | Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅) | ||
Theorem | oicl 8317 | The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ Ord dom 𝐹 | ||
Theorem | oif 8318 | The order isomorphism of the well-order 𝑅 on 𝐴 is a function. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ 𝐹:dom 𝐹⟶𝐴 | ||
Theorem | oiiso2 8319 | The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 8318). (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, ran 𝐹)) | ||
Theorem | ordtype 8320 | For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) | ||
Theorem | oiiniseg 8321 | ran 𝐹 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑁 ∈ 𝐴 ∧ 𝑀 ∈ dom 𝐹)) → ((𝐹‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝐹)) | ||
Theorem | ordtype2 8322 | For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto 𝐴 isomorphically. Otherwise, 𝐹 is a proper class, which implies that either ran 𝐹 ⊆ 𝐴 is a proper class or dom 𝐹 = On. This weak version of ordtype 8320 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) | ||
Theorem | oiexg 8323 | The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) | ||
Theorem | oion 8324 | The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On) | ||
Theorem | oiiso 8325 | The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) | ||
Theorem | oien 8326 | The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → dom 𝐹 ≈ 𝐴) | ||
Theorem | oieu 8327 | Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ((Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 (𝐵, 𝐴)) ↔ (𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹))) | ||
Theorem | oismo 8328 | When 𝐴 is a subclass of On, 𝐹 is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of 𝐴). The proof avoids ax-rep 4699 (the second statement is trivial under ax-rep 4699). (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = OrdIso( E , 𝐴) ⇒ ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴)) | ||
Theorem | oiid 8329 | The order type of an ordinal under the ∈ order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) | ||
Theorem | hartogslem1 8330* | Lemma for hartogs 8332. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} & ⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} ⇒ ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) | ||
Theorem | hartogslem2 8331* | Lemma for hartogs 8332. (Contributed by Mario Carneiro, 14-Jan-2013.) |
⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} & ⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) | ||
Theorem | hartogs 8332* | Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 9264- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) | ||
Theorem | wofib 8333 | The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ↔ (𝑅 We 𝐴 ∧ ◡𝑅 We 𝐴)) | ||
Theorem | wemaplem1 8334* | Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑄 ∈ 𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))) | ||
Theorem | wemaplem2 8335* | Lemma for wemapso 8339. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑𝑚 𝐴)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 𝐴)) & ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑𝑚 𝐴)) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑆 Po 𝐵) & ⊢ (𝜑 → 𝑎 ∈ 𝐴) & ⊢ (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎)) & ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) & ⊢ (𝜑 → 𝑏 ∈ 𝐴) & ⊢ (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏)) & ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) ⇒ ⊢ (𝜑 → 𝑃𝑇𝑄) | ||
Theorem | wemaplem3 8336* | Lemma for wemapso 8339. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑𝑚 𝐴)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 𝐴)) & ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑𝑚 𝐴)) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑆 Po 𝐵) & ⊢ (𝜑 → 𝑃𝑇𝑋) & ⊢ (𝜑 → 𝑋𝑇𝑄) ⇒ ⊢ (𝜑 → 𝑃𝑇𝑄) | ||
Theorem | wemappo 8337* |
Construct lexicographic order on a function space based on a
well-ordering of the indexes and a total ordering of the values.
Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑𝑚 𝐴)) | ||
Theorem | wemapsolem 8338* | Lemma for wemapso 8339. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 ⊆ (𝐵 ↑𝑚 𝐴) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑆 Or 𝐵) & ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) ∧ 𝑎 ≠ 𝑏)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐) ⇒ ⊢ (𝜑 → 𝑇 Or 𝑈) | ||
Theorem | wemapso 8339* | Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑𝑚 𝐴)) | ||
Theorem | wemapso2lem 8340* | Lemma for wemapso2 8341. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} ⇒ ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ 𝑊) → 𝑇 Or 𝑈) | ||
Theorem | wemapso2 8341* | An alternative to having a well-order on 𝑅 in wemapso 8339 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈) | ||
Theorem | card2on 8342* | Proof that the alternate definition cardval2 8700 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.) |
⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On | ||
Theorem | card2inf 8343* | The definition cardval2 8700 has the curious property that for non-numerable sets (for which ndmfv 6128 yields ∅), it still evaluates to a nonempty set, and indeed it contains ω. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) | ||
Syntax | char 8344 | Class symbol for the Hartogs/cardinal successor function. |
class har | ||
Syntax | cwdom 8345 | Class symbol for the weak dominance relation. |
class ≼* | ||
Definition | df-har 8346* |
Define the Hartogs function , which maps all sets to the smallest
ordinal that cannot be injected into the given set. In the important
special case where 𝑥 is an ordinal, this is the
cardinal successor
operation.
Traditionally, the Hartogs number of a set is written ℵ(𝑋) and the cardinal successor 𝑋 +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 8649. Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) | ||
Definition | df-wdom 8347* | A set is weakly dominated by a "larger" set iff the "larger" set can be mapped onto the "smaller" set or the smaller set is empty; equivalently if the smaller set can be placed into bijection with some partition of the larger set. When choice is assumed (as fodom 9227), this coincides with the 1-1 definition df-dom 7843; however, it is not known whether this is a choice-equivalent or a strictly weaker form. Some discussion of this question can be found at http://boolesrings.org/asafk/2014/on-the-partition-principle/. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ ≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} | ||
Theorem | harf 8348 | Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ har:V⟶On | ||
Theorem | harcl 8349 | Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (har‘𝑋) ∈ On | ||
Theorem | harval 8350* | Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}) | ||
Theorem | elharval 8351 | The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) | ||
Theorem | harndom 8352 | The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ ¬ (har‘𝑋) ≼ 𝑋 | ||
Theorem | harword 8353 | Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) ⊆ (har‘𝑌)) | ||
Theorem | relwdom 8354 | Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ Rel ≼* | ||
Theorem | brwdom 8355* | Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | ||
Theorem | brwdomi 8356* | Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)) | ||
Theorem | brwdomn0 8357* | Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≠ ∅ → (𝑋 ≼* 𝑌 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | ||
Theorem | 0wdom 8358 | Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → ∅ ≼* 𝑋) | ||
Theorem | fowdom 8359 | An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝑌–onto→𝑋) → 𝑋 ≼* 𝑌) | ||
Theorem | wdomref 8360 | Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → 𝑋 ≼* 𝑋) | ||
Theorem | brwdom2 8361* | Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ ∃𝑦 ∈ 𝒫 𝑌∃𝑧 𝑧:𝑦–onto→𝑋)) | ||
Theorem | domwdom 8362 | Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ≼ 𝑌 → 𝑋 ≼* 𝑌) | ||
Theorem | wdomtr 8363 | Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ ((𝑋 ≼* 𝑌 ∧ 𝑌 ≼* 𝑍) → 𝑋 ≼* 𝑍) | ||
Theorem | wdomen1 8364 | Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼* 𝐶 ↔ 𝐵 ≼* 𝐶)) | ||
Theorem | wdomen2 8365 | Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵)) | ||
Theorem | wdompwdom 8366 | Weak dominance strengthens to usual dominance on the power sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ≼* 𝑌 → 𝒫 𝑋 ≼ 𝒫 𝑌) | ||
Theorem | canthwdom 8367 | Cantor's Theorem, stated using weak dominance (this is actually a stronger statement than canth2 7998, equivalent to canth 6508). (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ¬ 𝒫 𝐴 ≼* 𝐴 | ||
Theorem | wdom2d 8368* | Deduce weak dominance from an implicit onto function (stated in a way which avoids ax-rep 4699). (Contributed by Stefan O'Rear, 13-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋) ⇒ ⊢ (𝜑 → 𝐴 ≼* 𝐵) | ||
Theorem | wdomd 8369* | Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.) |
⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑥 = 𝑋) ⇒ ⊢ (𝜑 → 𝐴 ≼* 𝐵) | ||
Theorem | brwdom3 8370* | Condition for weak dominance with a condition reminiscent of wdomd 8369. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑋 ≼* 𝑌 ↔ ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦))) | ||
Theorem | brwdom3i 8371* | Weak dominance implies existence of a covering function. (Contributed by Stefan O'Rear, 13-Feb-2015.) |
⊢ (𝑋 ≼* 𝑌 → ∃𝑓∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝑥 = (𝑓‘𝑦)) | ||
Theorem | unwdomg 8372 | Weak dominance of a (disjoint) union. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼* (𝐵 ∪ 𝐷)) | ||
Theorem | xpwdomg 8373 | Weak dominance of a Cartesian product. (Contributed by Stefan O'Rear, 13-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝐴 ≼* 𝐵 ∧ 𝐶 ≼* 𝐷) → (𝐴 × 𝐶) ≼* (𝐵 × 𝐷)) | ||
Theorem | wdomima2g 8374 | A set is weakly dominant over its image under any function. This version of wdomimag 8375 is stated so as to avoid ax-rep 4699. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉 ∧ (𝐹 “ 𝐴) ∈ 𝑊) → (𝐹 “ 𝐴) ≼* 𝐴) | ||
Theorem | wdomimag 8375 | A set is weakly dominant over its image under any function. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝑉) → (𝐹 “ 𝐴) ≼* 𝐴) | ||
Theorem | unxpwdom2 8376 | Lemma for unxpwdom 8377. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) | ||
Theorem | unxpwdom 8377 | If a Cartesian product is dominated by a union, then the base set is either weakly dominated by one factor of the union or dominated by the other. Extracted from Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((𝐴 × 𝐴) ≼ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) | ||
Theorem | harwdom 8378 | The Hartogs function is weakly dominated by 𝒫 (𝑋 × 𝑋). This follows from a more precise analysis of the bound used in hartogs 8332 to prove that (har‘𝑋) is a set. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) ≼* 𝒫 (𝑋 × 𝑋)) | ||
Theorem | ixpiunwdom 8379* | Describe an onto function from the indexed cartesian product to the indexed union. Together with ixpssmapg 7824 this shows that ∪ 𝑥 ∈ 𝐴𝐵 and X𝑥 ∈ 𝐴𝐵 have closely linked cardinalities. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 ∧ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼* (X𝑥 ∈ 𝐴 𝐵 × 𝐴)) | ||
Axiom | ax-reg 8380* | Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 8383) that every nonempty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 8387). A stronger version that works for proper classes is proved as zfregs 8491. (Contributed by NM, 14-Aug-1993.) |
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) | ||
Theorem | axreg2 8381* | Axiom of Regularity expressed more compactly. (Contributed by NM, 14-Aug-2003.) |
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | zfregcl 8382* | The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) |
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴)) | ||
Theorem | zfreg 8383* | The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." Axiom Reg of [BellMachover] p. 480. There is also a "strong form," not requiring that 𝐴 be a set, that can be proved with more difficulty (see zfregs 8491). (Contributed by NM, 26-Nov-1995.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
Theorem | zfregclOLD 8384* | Obsolete version of zfregcl 8382 as of 28-Apr-2021. (Contributed by NM, 5-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴) | ||
Theorem | zfregOLD 8385* | Obsolete version of zfreg 8383 as of 28-Apr-2021. (Contributed by NM, 26-Nov-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅) | ||
Theorem | zfreg2OLD 8386* | Alternate version of zfreg 8383 obsolete as of 28-Apr-2021. (Contributed by NM, 17-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝐴 ∩ 𝑥) = ∅) | ||
Theorem | elirrv 8387 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 8392 and efrirr 5019, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.) |
⊢ ¬ 𝑥 ∈ 𝑥 | ||
Theorem | elirr 8388 | No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
⊢ ¬ 𝐴 ∈ 𝐴 | ||
Theorem | sucprcreg 8389 | A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.) |
⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴) | ||
Theorem | ruv 8390 | The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | ||
Theorem | ruALT 8391 | Alternate proof of ru 3401, simplified using (indirectly) the Axiom of Regularity ax-reg 8380. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | zfregfr 8392 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
⊢ E Fr 𝐴 | ||
Theorem | en2lp 8393 | No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | ||
Theorem | en3lplem1 8394* | Lemma for en3lp 8396. (Contributed by Alan Sare, 28-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 = 𝐴 → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)) | ||
Theorem | en3lplem2 8395* | Lemma for en3lp 8396. (Contributed by Alan Sare, 28-Oct-2011.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝑥 ∩ {𝐴, 𝐵, 𝐶}) ≠ ∅)) | ||
Theorem | en3lp 8396 | No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 38102 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) | ||
Theorem | preleq 8397 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthreg 8398 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Regularity ax-reg 8380 (via the preleq 8397 step). See df-op 4132 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | suc11reg 8399 | The successor operation behaves like a one-to-one function (assuming the Axiom of Regularity). Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |
⊢ (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵) | ||
Theorem | dford2 8400* | Assuming ax-reg 8380, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))) |
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