Theorem bnj1177 34163

Description: Technical lemma for bnj69 34167. 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	|- ( ps <-> ( X e. B /\ y e. B /\ y R X ) )
bnj1177.3	|- C = ( trCl ( X , A , R ) i^i B )
bnj1177.9	|- ( ( ph /\ ps ) -> R FrSe A )
bnj1177.13	|- ( ( ph /\ ps ) -> B C_ A )
bnj1177.17	|- ( ( ph /\ ps ) -> X e. A )

Assertion

Ref	Expression
bnj1177	|- ( ( ph /\ ps ) -> ( R Fr A /\ C C_ A /\ C =/= (/) /\ C e. _V ) )

Proof of Theorem bnj1177

Step	Hyp	Ref	Expression
1		bnj1177.9	. . 3 |- ( ( ph /\ ps ) -> R FrSe A )
2		df-bnj15 33846	. . . 4 |- ( R FrSe A <-> ( R Fr A /\ R Se A ) )
3	2	simplbi 460	. . 3 |- ( R FrSe A -> R Fr A )
4	1, 3	syl 16	. 2 |- ( ( ph /\ ps ) -> R Fr A )
5		bnj1177.3	. . . 4 |- C = ( trCl ( X , A , R ) i^i B )
6		bnj1147 34151	. . . . 5 |- trCl ( X , A , R ) C_ A
7		ssinss1 3722	. . . . 5 |- ( trCl ( X , A , R ) C_ A -> ( trCl ( X , A , R ) i^i B ) C_ A )
8	6, 7	ax-mp 5	. . . 4 |- ( trCl ( X , A , R ) i^i B ) C_ A
9	5, 8	eqsstri 3529	. . 3 |- C C_ A
10	9	a1i 11	. 2 |- ( ( ph /\ ps ) -> C C_ A )
11		bnj1177.17	. . . . . . 7 |- ( ( ph /\ ps ) -> X e. A )
12		bnj906 34089	. . . . . . 7 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
13	1, 11, 12	syl2anc 661	. . . . . 6 |- ( ( ph /\ ps ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
14		ssrin 3719	. . . . . 6 |- ( pred ( X , A , R ) C_ trCl ( X , A , R ) -> ( pred ( X , A , R ) i^i B ) C_ ( trCl ( X , A , R ) i^i B ) )
15	13, 14	syl 16	. . . . 5 |- ( ( ph /\ ps ) -> ( pred ( X , A , R ) i^i B ) C_ ( trCl ( X , A , R ) i^i B ) )
16		bnj1177.13	. . . . . . . 8 |- ( ( ph /\ ps ) -> B C_ A )
17		bnj1177.2	. . . . . . . . . 10 |- ( ps <-> ( X e. B /\ y e. B /\ y R X ) )
18	17	simp2bi 1012	. . . . . . . . 9 |- ( ps -> y e. B )
19	18	adantl 466	. . . . . . . 8 |- ( ( ph /\ ps ) -> y e. B )
20	16, 19	sseldd 3500	. . . . . . 7 |- ( ( ph /\ ps ) -> y e. A )
21	17	simp3bi 1013	. . . . . . . 8 |- ( ps -> y R X )
22	21	adantl 466	. . . . . . 7 |- ( ( ph /\ ps ) -> y R X )
23		bnj1152 34155	. . . . . . 7 |- ( y e. pred ( X , A , R ) <-> ( y e. A /\ y R X ) )
24	20, 22, 23	sylanbrc 664	. . . . . 6 |- ( ( ph /\ ps ) -> y e. pred ( X , A , R ) )
25	24, 19	elind 3684	. . . . 5 |- ( ( ph /\ ps ) -> y e. ( pred ( X , A , R ) i^i B ) )
26	15, 25	sseldd 3500	. . . 4 |- ( ( ph /\ ps ) -> y e. ( trCl ( X , A , R ) i^i B ) )
27		ne0i 3799	. . . 4 |- ( y e. ( trCl ( X , A , R ) i^i B ) -> ( trCl ( X , A , R ) i^i B ) =/= (/) )
28	26, 27	syl 16	. . 3 |- ( ( ph /\ ps ) -> ( trCl ( X , A , R ) i^i B ) =/= (/) )
29	5	neeq1i 2742	. . 3 |- ( C =/= (/) <-> ( trCl ( X , A , R ) i^i B ) =/= (/) )
30	28, 29	sylibr 212	. 2 |- ( ( ph /\ ps ) -> C =/= (/) )
31		bnj893 34087	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) e. _V )
32	1, 11, 31	syl2anc 661	. . 3 |- ( ( ph /\ ps ) -> trCl ( X , A , R ) e. _V )
33		inex1g 4599	. . . 4 |- ( trCl ( X , A , R ) e. _V -> ( trCl ( X , A , R ) i^i B ) e. _V )
34	5, 33	syl5eqel 2549	. . 3 |- ( trCl ( X , A , R ) e. _V -> C e. _V )
35	32, 34	syl 16	. 2 |- ( ( ph /\ ps ) -> C e. _V )
36	4, 10, 30, 35	bnj951 33935	1 |- ( ( ph /\ ps ) -> ( R Fr A /\ C C_ A /\ C =/= (/) /\ C e. _V ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   _Vcvv 3109   i^i cin 3470   C_ wss 3471   (/)c0 3793   class class class wbr 4456   Fr wfr 4844   /\ w-bnj17 33839   predc-bnj14 33841   Se w-bnj13 33843   FrSe w-bnj15 33845   trClc-bnj18 33847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-bnj17 33840  df-bnj14 33842  df-bnj13 33844  df-bnj15 33846  df-bnj18 33848

This theorem is referenced by:  bnj1190  34165