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Theorem bnj1190 31354
 Description: Technical lemma for bnj69 31356. 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 1138 . . . . . 6 (𝜑𝐵𝐴)
32adantr 472 . . . . 5 ((𝜑𝜓) → 𝐵𝐴)
4 eqid 2748 . . . . . 6 ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5 bnj1190.2 . . . . . . . . 9 (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))
61simp1bi 1137 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
76adantr 472 . . . . . . . . 9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
85simp1bi 1137 . . . . . . . . . 10 (𝜓𝑥𝐵)
9 ssel2 3727 . . . . . . . . . 10 ((𝐵𝐴𝑥𝐵) → 𝑥𝐴)
102, 8, 9syl2an 495 . . . . . . . . 9 ((𝜑𝜓) → 𝑥𝐴)
115, 4, 7, 3, 10bnj1177 31352 . . . . . . . 8 ((𝜑𝜓) → (𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V))
12 bnj1154 31345 . . . . . . . 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 31351 . . . . . . 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 251 . . . . . . . 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 251 . . . . . . . 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 31350 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢𝑣 ∈ trCl(𝑥, 𝐴, 𝑅)))
174, 13, 16bnj1174 31349 . . . . . 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 31348 . . . . . 6 ((𝜑𝜓𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) → (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) ↔ 𝑣𝐴))
194, 17, 18bnj1172 31347 . . . . 5 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣𝐵))))
203, 19bnj1171 31346 . . . 4 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐵 → ¬ 𝑣𝑅𝑢)))
2120bnj1186 31353 . . 3 ((𝜑𝜓) → ∃𝑢𝐵𝑣𝐵 ¬ 𝑣𝑅𝑢)
2221bnj1185 31142 . 2 ((𝜑𝜓) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2322bnj1185 31142 1 ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   ∈ wcel 2127   ≠ wne 2920  ∀wral 3038  ∃wrex 3039   ∩ cin 3702   ⊆ wss 3703  ∅c0 4046   class class class wbr 4792   FrSe w-bnj15 31038   trClc-bnj18 31040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-reg 8650  ax-inf2 8699 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-om 7219  df-1o 7717  df-bnj17 31033  df-bnj14 31035  df-bnj13 31037  df-bnj15 31039  df-bnj18 31041  df-bnj19 31043 This theorem is referenced by:  bnj1189  31355
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