Theorem bnj1175 35500

Description: Technical lemma for bnj69 35506. 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 35202	. . . . 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 35184	. . . . 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 302	. . . 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 636	. . . 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 281	. . 3 ⊢ (𝜒 ↔ (𝜃 ∧ 𝑤𝑅𝑧))
8		bnj1125 35488	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
9	1, 8	bnj835 35256	. . . 4 ⊢ (𝜒 → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
10		bnj906 35426	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
11	1, 10	bnj836 35257	. . . . 5 ⊢ (𝜒 → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
12		bnj1152 35494	. . . . . . 7 ⊢ (𝑤 ∈ pred(𝑧, 𝐴, 𝑅) ↔ (𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧))
13	12	biimpri 231	. . . . . 6 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ pred(𝑧, 𝐴, 𝑅))
14	1, 13	bnj837 35258	. . . . 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 238	. 2 ⊢ ((𝜃 ∧ 𝑤𝑅𝑧) → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅))
18	17	ex 418	1 ⊢ (𝜃 → (𝑤𝑅𝑧 → 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5107   ∧ w-bnj17 35183   predc-bnj14 35185   FrSe w-bnj15 35189   trClc-bnj18 35191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-reg 9567  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7866  df-1o 8458  df-bnj17 35184  df-bnj14 35186  df-bnj13 35188  df-bnj15 35190  df-bnj18 35192  df-bnj19 35194

This theorem is used by:  bnj1190  35504