Theorem bnj1177 32271

Description: Technical lemma for bnj69 32275. 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 31956	. . . 4 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
3	2	simplbi 500	. . 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 32259	. . . . 5 ⊢ trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7		ssinss1 4212	. . . . 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 3999	. . 3 ⊢ 𝐶 ⊆ 𝐴
10	9	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐴)
11		bnj1177.17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐴)
12		bnj906 32195	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
13	1, 11, 12	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → pred(𝑋, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
14	13	ssrind 4210	. . . . 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 1141	. . . . . . . . 9 ⊢ (𝜓 → 𝑦 ∈ 𝐵)
18	17	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)
19	15, 18	sseldd 3966	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐴)
20	16	simp3bi 1142	. . . . . . . 8 ⊢ (𝜓 → 𝑦𝑅𝑋)
21	20	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝑦𝑅𝑋)
22		bnj1152 32263	. . . . . . 7 ⊢ (𝑦 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑋))
23	19, 21, 22	sylanbrc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ pred(𝑋, 𝐴, 𝑅))
24	23, 18	elind 4169	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( pred(𝑋, 𝐴, 𝑅) ∩ 𝐵))
25	14, 24	sseldd 3966	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
26	25	ne0d 4299	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
27	5	neeq1i 3078	. . 3 ⊢ (𝐶 ≠ ∅ ↔ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ≠ ∅)
28	26, 27	sylibr 236	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≠ ∅)
29		bnj893 32193	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
30	1, 11, 29	syl2anc 586	. . 3 ⊢ ((𝜑 ∧ 𝜓) → trCl(𝑋, 𝐴, 𝑅) ∈ V)
31		inex1g 5214	. . . 4 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ∈ V)
32	5, 31	eqeltrid 2915	. . 3 ⊢ ( trCl(𝑋, 𝐴, 𝑅) ∈ V → 𝐶 ∈ V)
33	30, 32	syl 17	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ V)
34	4, 10, 28, 33	bnj951 32040	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 Fr 𝐴 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ ∅ ∧ 𝐶 ∈ V))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ∅c0 4289   class class class wbr 5057   Fr wfr 5504   ∧ w-bnj17 31949   predc-bnj14 31951   Se w-bnj13 31953   FrSe w-bnj15 31955   trClc-bnj18 31957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-reg 9048  ax-inf2 9096

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-1o 8094  df-bnj17 31950  df-bnj14 31952  df-bnj13 31954  df-bnj15 31956  df-bnj18 31958

This theorem is referenced by:  bnj1190  32273