Theorem bnj1177 29887

Description: Technical lemma for bnj69 29891. 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 29570	. . . 4 |- ( R FrSe A <-> ( R Fr A /\ R Se A ) )
3	2	simplbi 467	. . 3 |- ( R FrSe A -> R Fr A )
4	1, 3	syl 17	. 2 |- ( ( ph /\ ps ) -> R Fr A )
5		bnj1177.3	. . . 4 |- C = ( trCl ( X , A , R ) i^i B )
6		bnj1147 29875	. . . . 5 |- trCl ( X , A , R ) C_ A
7		ssinss1 3651	. . . . 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 3448	. . 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 29813	. . . . . . 7 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
13	1, 11, 12	syl2anc 673	. . . . . 6 |- ( ( ph /\ ps ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
14		ssrin 3648	. . . . . 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 17	. . . . 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 1046	. . . . . . . . 9 |- ( ps -> y e. B )
19	18	adantl 473	. . . . . . . 8 |- ( ( ph /\ ps ) -> y e. B )
20	16, 19	sseldd 3419	. . . . . . 7 |- ( ( ph /\ ps ) -> y e. A )
21	17	simp3bi 1047	. . . . . . . 8 |- ( ps -> y R X )
22	21	adantl 473	. . . . . . 7 |- ( ( ph /\ ps ) -> y R X )
23		bnj1152 29879	. . . . . . 7 |- ( y e. pred ( X , A , R ) <-> ( y e. A /\ y R X ) )
24	20, 22, 23	sylanbrc 677	. . . . . 6 |- ( ( ph /\ ps ) -> y e. pred ( X , A , R ) )
25	24, 19	elind 3609	. . . . 5 |- ( ( ph /\ ps ) -> y e. ( pred ( X , A , R ) i^i B ) )
26	15, 25	sseldd 3419	. . . 4 |- ( ( ph /\ ps ) -> y e. ( trCl ( X , A , R ) i^i B ) )
27		ne0i 3728	. . . 4 |- ( y e. ( trCl ( X , A , R ) i^i B ) -> ( trCl ( X , A , R ) i^i B ) =/= (/) )
28	26, 27	syl 17	. . 3 |- ( ( ph /\ ps ) -> ( trCl ( X , A , R ) i^i B ) =/= (/) )
29	5	neeq1i 2707	. . 3 |- ( C =/= (/) <-> ( trCl ( X , A , R ) i^i B ) =/= (/) )
30	28, 29	sylibr 217	. 2 |- ( ( ph /\ ps ) -> C =/= (/) )
31		bnj893 29811	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) e. _V )
32	1, 11, 31	syl2anc 673	. . 3 |- ( ( ph /\ ps ) -> trCl ( X , A , R ) e. _V )
33		inex1g 4539	. . . 4 |- ( trCl ( X , A , R ) e. _V -> ( trCl ( X , A , R ) i^i B ) e. _V )
34	5, 33	syl5eqel 2553	. . 3 |- ( trCl ( X , A , R ) e. _V -> C e. _V )
35	32, 34	syl 17	. 2 |- ( ( ph /\ ps ) -> C e. _V )
36	4, 10, 30, 35	bnj951 29659	1 |- ( ( ph /\ ps ) -> ( R Fr A /\ C C_ A /\ C =/= (/) /\ C e. _V ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   _Vcvv 3031   i^i cin 3389   C_ wss 3390   (/)c0 3722   class class class wbr 4395   Fr wfr 4795   /\ w-bnj17 29563   predc-bnj14 29565   Se w-bnj13 29567   FrSe w-bnj15 29569   trClc-bnj18 29571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-bnj17 29564  df-bnj14 29566  df-bnj13 29568  df-bnj15 29570  df-bnj18 29572

This theorem is referenced by:  bnj1190  29889