Theorem bnj1177 31994

Description: Technical lemma for bnj69 31998. 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 31678	. . . 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 31982	. . . . 5 |- trCl ( X , A , R ) C_ A
7		ssinss1 3576	. . . . 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 3384	. . 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 31920	. . . . . . 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 3573	. . . . . 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 1004	. . . . . . . . 9 |- ( ps -> y e. B )
19	18	adantl 466	. . . . . . . 8 |- ( ( ph /\ ps ) -> y e. B )
20	16, 19	sseldd 3355	. . . . . . 7 |- ( ( ph /\ ps ) -> y e. A )
21	17	simp3bi 1005	. . . . . . . 8 |- ( ps -> y R X )
22	21	adantl 466	. . . . . . 7 |- ( ( ph /\ ps ) -> y R X )
23		bnj1152 31986	. . . . . . 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 3538	. . . . 5 |- ( ( ph /\ ps ) -> y e. ( pred ( X , A , R ) i^i B ) )
26	15, 25	sseldd 3355	. . . 4 |- ( ( ph /\ ps ) -> y e. ( trCl ( X , A , R ) i^i B ) )
27		ne0i 3641	. . . 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 2616	. . 3 |- ( C =/= (/) <-> ( trCl ( X , A , R ) i^i B ) =/= (/) )
30	28, 29	sylibr 212	. 2 |- ( ( ph /\ ps ) -> C =/= (/) )
31		bnj893 31918	. . . 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 4433	. . . 4 |- ( trCl ( X , A , R ) e. _V -> ( trCl ( X , A , R ) i^i B ) e. _V )
34	5, 33	syl5eqel 2525	. . 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 31766	1 |- ( ( ph /\ ps ) -> ( R Fr A /\ C C_ A /\ C =/= (/) /\ C e. _V ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2604   _Vcvv 2970   i^i cin 3325   C_ wss 3326   (/)c0 3635   class class class wbr 4290   Fr wfr 4674   /\ w-bnj17 31671   predc-bnj14 31673   Se w-bnj13 31675   FrSe w-bnj15 31677   trClc-bnj18 31679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-reg 7805  ax-inf2 7845

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-om 6475  df-1o 6918  df-bnj17 31672  df-bnj14 31674  df-bnj13 31676  df-bnj15 31678  df-bnj18 31680

This theorem is referenced by:  bnj1190  31996