Theorem bnj1173 32269

Description: Technical lemma for bnj69 32277. 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 1146	. . . 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 1128	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑅 FrSe 𝐴)
5		bnj1173.17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)
6	5	3adant3 1128	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑋 ∈ 𝐴)
7		elin 4168	. . . . . . . . 9 ⊢ (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ 𝐵))
8	7	simplbi 500	. . . . . . . 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 2931	. . . . . . 7 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
11	10	3ad2ant3 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
12		pm3.21 474	. . . . . 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 1367	. . . . 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 31963	. . . . 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 254	. . . 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 228	. . 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	syl5bb 285	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)))
18		bnj1147 32261	. . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
19	18, 11	bnj1213 32065	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
20	4, 19	jca 514	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴))
21	20	biantrurd 535	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝑤 ∈ 𝐴 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ 𝑤 ∈ 𝐴)))
22	17, 21	bitr4d 284	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ 𝐶) → (𝜃 ↔ 𝑤 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ∩ cin 3934   FrSe w-bnj15 31957   trClc-bnj18 31959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fn 6352  df-fv 6357  df-om 7575  df-bnj17 31952  df-bnj14 31954  df-bnj18 31960

This theorem is referenced by:  bnj1190  32275