Theorem bnj1175 35162

Description: Technical lemma for bnj69 35168. 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 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧)))
bnj1175.5	⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))

Assertion

Ref	Expression
bnj1175	⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))

Proof of Theorem bnj1175

Step	Hyp	Ref	Expression
1		bnj1175.4	. . . . 5 ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧)))
2		bnj255 34863	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧)))
3		df-bnj17 34845	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤𝑅𝑧))
4	1, 2, 3	3bitr2i 299	. . . 4 ⊢ (𝜒 ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤𝑅𝑧))
5		bnj1175.5	. . . . 5 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))
6	5	anbi1i 625	. . . 4 ⊢ ((𝜃 ∧ 𝑤𝑅𝑧) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤𝑅𝑧))
7	4, 6	bitr4i 278	. . 3 ⊢ (𝜒 ↔ (𝜃 ∧ 𝑤𝑅𝑧))
8		bnj1125 35150	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
9	1, 8	bnj835 34917	. . . 4 ⊢ (𝜒 → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
10		bnj906 35088	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
11	1, 10	bnj836 34918	. . . . 5 ⊢ (𝜒 → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
12		bnj1152 35156	. . . . . . 7 ⊢ (𝑤 ∈ pred(𝑧, 𝐴, 𝑅) ↔ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧))
13	12	biimpri 228	. . . . . 6 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅))
14	1, 13	bnj837 34919	. . . . 5 ⊢ (𝜒 → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅))
15	11, 14	sseldd 3935	. . . 4 ⊢ (𝜒 → 𝑤 ∈ trCl(𝑧, 𝐴, 𝑅))
16	9, 15	sseldd 3935	. . 3 ⊢ (𝜒 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))
17	7, 16	sylbir 235	. 2 ⊢ ((𝜃 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))
18	17	ex 412	1 ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5099   ∧ w-bnj17 34844   predc-bnj14 34846   FrSe w-bnj15 34850   trClc-bnj18 34852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-reg 9501  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7811  df-1o 8399  df-bnj17 34845  df-bnj14 34847  df-bnj13 34849  df-bnj15 34851  df-bnj18 34853  df-bnj19 34855

This theorem is referenced by:  bnj1190  35166