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Theorem bnj1190 30330
Description: 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.)
Hypotheses
Ref Expression
bnj1190.1 (𝜑 ↔ (𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅))
bnj1190.2 (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))
Assertion
Ref Expression
bnj1190 ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
Distinct variable groups:   𝑤,𝐵,𝑥,𝑧   𝑦,𝐵,𝑥,𝑧   𝑤,𝑅,𝑥,𝑧   𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem bnj1190
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1190.1 . . . . . . 7 (𝜑 ↔ (𝑅 FrSe 𝐴𝐵𝐴𝐵 ≠ ∅))
21simp2bi 1070 . . . . . 6 (𝜑𝐵𝐴)
32adantr 480 . . . . 5 ((𝜑𝜓) → 𝐵𝐴)
4 eqid 2610 . . . . . 6 ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5 bnj1190.2 . . . . . . . . 9 (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))
61simp1bi 1069 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
76adantr 480 . . . . . . . . 9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
85simp1bi 1069 . . . . . . . . . 10 (𝜓𝑥𝐵)
9 ssel2 3563 . . . . . . . . . 10 ((𝐵𝐴𝑥𝐵) → 𝑥𝐴)
102, 8, 9syl2an 493 . . . . . . . . 9 ((𝜑𝜓) → 𝑥𝐴)
115, 4, 7, 3, 10bnj1177 30328 . . . . . . . 8 ((𝜑𝜓) → (𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V))
12 bnj1154 30321 . . . . . . . 8 ((𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V) → ∃𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)∀𝑣 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ¬ 𝑣𝑅𝑢)
1311, 12bnj1176 30327 . . . . . . 7 𝑢𝑣((𝜑𝜓) → (𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∧ (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢 → ¬ 𝑣 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)))))
14 biid 250 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ (𝑣𝐴𝑣𝑅𝑢)) ↔ ((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ (𝑣𝐴𝑣𝑅𝑢)))
15 biid 250 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) ↔ ((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴))
164, 14, 15bnj1175 30326 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢𝑣 ∈ trCl(𝑥, 𝐴, 𝑅)))
174, 13, 16bnj1174 30325 . . . . . 6 𝑢𝑣((𝜑𝜓) → ((𝜑𝜓𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) ∧ (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢 → ¬ 𝑣𝐵))))
184, 15, 7, 10bnj1173 30324 . . . . . 6 ((𝜑𝜓𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) → (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) ↔ 𝑣𝐴))
194, 17, 18bnj1172 30323 . . . . 5 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣𝐵))))
203, 19bnj1171 30322 . . . 4 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐵 → ¬ 𝑣𝑅𝑢)))
2120bnj1186 30329 . . 3 ((𝜑𝜓) → ∃𝑢𝐵𝑣𝐵 ¬ 𝑣𝑅𝑢)
2221bnj1185 30118 . 2 ((𝜑𝜓) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2322bnj1185 30118 1 ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031  wcel 1977  wne 2780  wral 2896  wrex 2897  cin 3539  wss 3540  c0 3874   class class class wbr 4583   FrSe w-bnj15 30011   trClc-bnj18 30013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016
This theorem is referenced by:  bnj1189  30331
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