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Theorem bnj1190 32996
Description: Technical lemma for bnj69 32998. 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 1145 . . . . . 6 (𝜑𝐵𝐴)
32adantr 481 . . . . 5 ((𝜑𝜓) → 𝐵𝐴)
4 eqid 2738 . . . . . 6 ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5 bnj1190.2 . . . . . . . . 9 (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))
61simp1bi 1144 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
76adantr 481 . . . . . . . . 9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
85simp1bi 1144 . . . . . . . . . 10 (𝜓𝑥𝐵)
9 ssel2 3915 . . . . . . . . . 10 ((𝐵𝐴𝑥𝐵) → 𝑥𝐴)
102, 8, 9syl2an 596 . . . . . . . . 9 ((𝜑𝜓) → 𝑥𝐴)
115, 4, 7, 3, 10bnj1177 32994 . . . . . . . 8 ((𝜑𝜓) → (𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V))
12 bnj1154 32987 . . . . . . . 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 32993 . . . . . . 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 260 . . . . . . . 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 260 . . . . . . . 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 32992 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢𝑣 ∈ trCl(𝑥, 𝐴, 𝑅)))
174, 13, 16bnj1174 32991 . . . . . 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 32990 . . . . . 6 ((𝜑𝜓𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) → (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) ↔ 𝑣𝐴))
194, 17, 18bnj1172 32989 . . . . 5 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣𝐵))))
203, 19bnj1171 32988 . . . 4 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐵 → ¬ 𝑣𝑅𝑢)))
2120bnj1186 32995 . . 3 ((𝜑𝜓) → ∃𝑢𝐵𝑣𝐵 ¬ 𝑣𝑅𝑢)
2221bnj1185 32781 . 2 ((𝜑𝜓) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2322bnj1185 32781 1 ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086  wcel 2106  wne 2943  wral 3064  wrex 3065  cin 3885  wss 3886  c0 4256   class class class wbr 5073   FrSe w-bnj15 32679   trClc-bnj18 32681
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-reg 9338  ax-inf2 9386
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-om 7703  df-1o 8284  df-bnj17 32674  df-bnj14 32676  df-bnj13 32678  df-bnj15 32680  df-bnj18 32682  df-bnj19 32684
This theorem is referenced by:  bnj1189  32997
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