Theorem bnj1177 34989

Description: Technical lemma for bnj69 34993. 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 34676	. . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
3	2	simplbi 497	. . 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 34977	. . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7		ssinss1 4205	. . . . 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 3990	. . 3 ⊢ 𝐶 ⊆ 𝐴
10	9	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴)
11		bnj1177.17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)
12		bnj906 34913	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
13	1, 11, 12	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
14	13	ssrind 4203	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵) ⊆ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
15		bnj1177.13	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ⊆ 𝐴)
16		bnj1177.2	. . . . . . . . . 10 ⊢ (𝜓 ↔ (𝑋 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦𝑅𝑋))
17	16	simp2bi 1146	. . . . . . . . 9 ⊢ (𝜓 → 𝑦 ∈ 𝐵)
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)
19	15, 18	sseldd 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴)
20	16	simp3bi 1147	. . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋)
21	20	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋)
22		bnj1152 34981	. . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋))
23	19, 21, 22	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
24	23, 18	elind 4159	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
25	14, 24	sseldd 3944	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
26	25	ne0d 4301	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
27	5	neeq1i 2989	. . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
28	26, 27	sylibr 234	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅)
29		bnj893 34911	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
30	1, 11, 29	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
31		inex1g 5269	. . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
32	5, 31	eqeltrid 2832	. . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
33	30, 32	syl 17	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V)
34	4, 10, 28, 33	bnj951 34758	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102   Fr wfr 5581   ∧ w-bnj17 34669   predc-bnj14 34671   Se w-bnj13 34673   FrSe w-bnj15 34675   trClc-bnj18 34677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-bnj17 34670  df-bnj14 34672  df-bnj13 34674  df-bnj15 34676  df-bnj18 34678

This theorem is referenced by:  bnj1190  34991