Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1190 Structured version   Visualization version   GIF version

Theorem bnj1190 34672
Description: Technical lemma for bnj69 34674. 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 1143 . . . . . 6 (𝜑𝐵𝐴)
32adantr 479 . . . . 5 ((𝜑𝜓) → 𝐵𝐴)
4 eqid 2728 . . . . . 6 ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) = ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)
5 bnj1190.2 . . . . . . . . 9 (𝜓 ↔ (𝑥𝐵𝑦𝐵𝑦𝑅𝑥))
61simp1bi 1142 . . . . . . . . . 10 (𝜑𝑅 FrSe 𝐴)
76adantr 479 . . . . . . . . 9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
85simp1bi 1142 . . . . . . . . . 10 (𝜓𝑥𝐵)
9 ssel2 3977 . . . . . . . . . 10 ((𝐵𝐴𝑥𝐵) → 𝑥𝐴)
102, 8, 9syl2an 594 . . . . . . . . 9 ((𝜑𝜓) → 𝑥𝐴)
115, 4, 7, 3, 10bnj1177 34670 . . . . . . . 8 ((𝜑𝜓) → (𝑅 Fr 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴 ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅ ∧ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵) ∈ V))
12 bnj1154 34663 . . . . . . . 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 34669 . . . . . . 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 34668 . . . . . . 7 (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) → (𝑣𝑅𝑢𝑣 ∈ trCl(𝑥, 𝐴, 𝑅)))
174, 13, 16bnj1174 34667 . . . . . 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 34666 . . . . . 6 ((𝜑𝜓𝑢 ∈ ( trCl(𝑥, 𝐴, 𝑅) ∩ 𝐵)) → (((𝑅 FrSe 𝐴𝑥𝐴𝑢 ∈ trCl(𝑥, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑢𝐴) ∧ 𝑣𝐴) ↔ 𝑣𝐴))
194, 17, 18bnj1172 34665 . . . . 5 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐴 → (𝑣𝑅𝑢 → ¬ 𝑣𝐵))))
203, 19bnj1171 34664 . . . 4 𝑢𝑣((𝜑𝜓) → (𝑢𝐵 ∧ (𝑣𝐵 → ¬ 𝑣𝑅𝑢)))
2120bnj1186 34671 . . 3 ((𝜑𝜓) → ∃𝑢𝐵𝑣𝐵 ¬ 𝑣𝑅𝑢)
2221bnj1185 34457 . 2 ((𝜑𝜓) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
2322bnj1185 34457 1 ((𝜑𝜓) → ∃𝑤𝐵𝑧𝐵 ¬ 𝑧𝑅𝑤)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084  wcel 2098  wne 2937  wral 3058  wrex 3067  cin 3948  wss 3949  c0 4326   class class class wbr 5152   FrSe w-bnj15 34356   trClc-bnj18 34358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9623  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7877  df-1o 8493  df-bnj17 34351  df-bnj14 34353  df-bnj13 34355  df-bnj15 34357  df-bnj18 34359  df-bnj19 34361
This theorem is referenced by:  bnj1189  34673
  Copyright terms: Public domain W3C validator