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Theorem bnj1190 33677
Description: Technical lemma for bnj69 33679. 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 1147 . . . . . 6 (𝜑𝐵𝐴)
32adantr 482 . . . . 5 ((𝜑𝜓) → 𝐵𝐴)
4 eqid 2733 . . . . . 6 ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5 bnj1190.2 . . . . . . . . 9 (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))
61simp1bi 1146 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
76adantr 482 . . . . . . . . 9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
85simp1bi 1146 . . . . . . . . . 10 (𝜓𝑥𝐵)
9 ssel2 3940 . . . . . . . . . 10 ((𝐵𝐴𝑥𝐵) → 𝑥𝐴)
102, 8, 9syl2an 597 . . . . . . . . 9 ((𝜑𝜓) → 𝑥𝐴)
115, 4, 7, 3, 10bnj1177 33675 . . . . . . . 8 ((𝜑𝜓) → (𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V))
12 bnj1154 33668 . . . . . . . 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 33674 . . . . . . 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 261 . . . . . . . 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 261 . . . . . . . 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 33673 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢𝑣 ∈ trCl(𝑥, 𝐴, 𝑅)))
174, 13, 16bnj1174 33672 . . . . . 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 33671 . . . . . 6 ((𝜑𝜓𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) → (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) ↔ 𝑣𝐴))
194, 17, 18bnj1172 33670 . . . . 5 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣𝐵))))
203, 19bnj1171 33669 . . . 4 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐵 → ¬ 𝑣𝑅𝑢)))
2120bnj1186 33676 . . 3 ((𝜑𝜓) → ∃𝑢𝐵𝑣𝐵 ¬ 𝑣𝑅𝑢)
2221bnj1185 33462 . 2 ((𝜑𝜓) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2322bnj1185 33462 1 ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088  wcel 2107  wne 2940  wral 3061  wrex 3070  cin 3910  wss 3911  c0 4283   class class class wbr 5106   FrSe w-bnj15 33361   trClc-bnj18 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804  df-1o 8413  df-bnj17 33356  df-bnj14 33358  df-bnj13 33360  df-bnj15 33362  df-bnj18 33364  df-bnj19 33366
This theorem is referenced by:  bnj1189  33678
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