Theorem bnj1173 33222

Description: Technical lemma for bnj69 33230. 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
bnj1173.3	⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1173.5	⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))
bnj1173.9	⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
bnj1173.17	⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
bnj1173	⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))

Proof of Theorem bnj1173

Step	Hyp	Ref	Expression
1		bnj1173.5	. . 3 ⊢ (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))
2		3simpc 1149	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴))
3		bnj1173.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
4	3	3adant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑅 FrSe 𝐴)
5		bnj1173.17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)
6	5	3adant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑋 ∈ 𝐴)
7		elin 3913	. . . . . . . . 9 ⊢ (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ 𝐵))
8	7	simplbi 498	. . . . . . . 8 ⊢ (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
9		bnj1173.3	. . . . . . . 8 ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
10	8, 9	eleq2s 2855	. . . . . . 7 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
11	10	3ad2ant3 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
12		pm3.21 472	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → (((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → (((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
13	4, 6, 11, 12	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → (((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
14		bnj170 32918	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ∧ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
15	13, 14	syl6ibr 251	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)))
16	2, 15	impbid2 225	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)))
17	1, 16	bitrid 282	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)))
18		bnj1147 33214	. . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
19	18, 11	bnj1213 33018	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
20	4, 19	jca 512	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴))
21	20	biantrurd 533	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝑤 ∈ 𝐴 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)))
22	17, 21	bitr4d 281	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ∩ cin 3896   FrSe w-bnj15 32912   trClc-bnj18 32914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-tr 5207  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fn 6476  df-fv 6481  df-om 7773  df-bnj17 32907  df-bnj14 32909  df-bnj18 32915

This theorem is referenced by:  bnj1190  33228