Theorem bnj1177 30328

Description: Technical lemma for bnj69 30332. 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
bnj1177.2	⊢ (𝜓 ↔ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑋))
bnj1177.3	⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1177.9	⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
bnj1177.13	⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
bnj1177.17	⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
bnj1177	⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V))

Proof of Theorem bnj1177

Step	Hyp	Ref	Expression
1		bnj1177.9	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 FrSe 𝐴)
2		df-bnj15 30012	. . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
3	2	simplbi 475	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Fr 𝐴)
4	1, 3	syl 17	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 Fr 𝐴)
5		bnj1177.3	. . . 4 ⊢ 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
6		bnj1147 30316	. . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7		ssinss1 3803	. . . . 5 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴 → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴)
8	6, 7	ax-mp 5	. . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ 𝐴
9	5, 8	eqsstri 3598	. . 3 ⊢ 𝐶 ⊆ 𝐴
10	9	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴)
11		bnj1177.17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)
12		bnj906 30254	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
13	1, 11, 12	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
14		ssrin 3800	. . . . . 6 ⊢ ( pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
15	13, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
16		bnj1177.13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
17		bnj1177.2	. . . . . . . . . 10 ⊢ (𝜓 ↔ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑋))
18	17	simp2bi 1070	. . . . . . . . 9 ⊢ (𝜓 → 𝑦 ∈ 𝐵)
19	18	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)
20	16, 19	sseldd 3569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴)
21	17	simp3bi 1071	. . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋)
22	21	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋)
23		bnj1152 30320	. . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋))
24	20, 22, 23	sylanbrc 695	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
25	24, 19	elind 3760	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
26	15, 25	sseldd 3569	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
27		ne0i 3880	. . . 4 ⊢ (𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
28	26, 27	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
29	5	neeq1i 2846	. . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
30	28, 29	sylibr 223	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅)
31		bnj893 30252	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
32	1, 11, 31	syl2anc 691	. . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
33		inex1g 4729	. . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
34	5, 33	syl5eqel 2692	. . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
35	32, 34	syl 17	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V)
36	4, 10, 30, 35	bnj951 30100	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   Fr wfr 4994   ∧ w-bnj17 30005   predc-bnj14 30007   Se w-bnj13 30009   FrSe w-bnj15 30011   trClc-bnj18 30013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014

This theorem is referenced by:  bnj1190  30330