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Theorem bnj1175 30777
Description: Technical lemma for bnj69 30783. 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
bnj1175.3 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1175.4 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ (𝑤𝐴𝑤𝑅𝑧)))
bnj1175.5 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))
Assertion
Ref Expression
bnj1175 (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))

Proof of Theorem bnj1175
StepHypRef Expression
1 bnj1175.4 . . . . 5 (𝜒 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ (𝑤𝐴𝑤𝑅𝑧)))
2 bnj255 30475 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴𝑤𝑅𝑧) ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ (𝑤𝐴𝑤𝑅𝑧)))
3 df-bnj17 30457 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴𝑤𝑅𝑧) ↔ (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ 𝑤𝑅𝑧))
41, 2, 33bitr2i 288 . . . 4 (𝜒 ↔ (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ 𝑤𝑅𝑧))
5 bnj1175.5 . . . . 5 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))
65anbi1i 730 . . . 4 ((𝜃𝑤𝑅𝑧) ↔ (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ 𝑤𝑅𝑧))
74, 6bitr4i 267 . . 3 (𝜒 ↔ (𝜃𝑤𝑅𝑧))
8 bnj1125 30765 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
91, 8bnj835 30534 . . . 4 (𝜒 → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
10 bnj906 30705 . . . . . 6 ((𝑅 FrSe 𝐴𝑧𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
111, 10bnj836 30535 . . . . 5 (𝜒 → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
12 bnj1152 30771 . . . . . . 7 (𝑤 ∈ pred(𝑧, 𝐴, 𝑅) ↔ (𝑤𝐴𝑤𝑅𝑧))
1312biimpri 218 . . . . . 6 ((𝑤𝐴𝑤𝑅𝑧) → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅))
141, 13bnj837 30536 . . . . 5 (𝜒𝑤 ∈ pred(𝑧, 𝐴, 𝑅))
1511, 14sseldd 3584 . . . 4 (𝜒𝑤 ∈ trCl(𝑧, 𝐴, 𝑅))
169, 15sseldd 3584 . . 3 (𝜒𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))
177, 16sylbir 225 . 2 ((𝜃𝑤𝑅𝑧) → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))
1817ex 450 1 (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cin 3554  wss 3555   class class class wbr 4613  w-bnj17 30456   predc-bnj14 30458   FrSe w-bnj15 30462   trClc-bnj18 30464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-bnj17 30457  df-bnj14 30459  df-bnj13 30461  df-bnj15 30463  df-bnj18 30465  df-bnj19 30467
This theorem is referenced by:  bnj1190  30781
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