Theorem bnj1177 29389

Description: Technical lemma for bnj69 29393. 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 29072	. . . 4 |- ( R FrSe A <-> ( R Fr A /\ R Se A ) )
3	2	simplbi 458	. . 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 29377	. . . . 5 |- trCl ( X , A , R ) C_ A
7		ssinss1 3667	. . . . 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 3472	. . 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 29315	. . . . . . 7 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
13	1, 11, 12	syl2anc 659	. . . . . 6 |- ( ( ph /\ ps ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
14		ssrin 3664	. . . . . 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 1013	. . . . . . . . 9 |- ( ps -> y e. B )
19	18	adantl 464	. . . . . . . 8 |- ( ( ph /\ ps ) -> y e. B )
20	16, 19	sseldd 3443	. . . . . . 7 |- ( ( ph /\ ps ) -> y e. A )
21	17	simp3bi 1014	. . . . . . . 8 |- ( ps -> y R X )
22	21	adantl 464	. . . . . . 7 |- ( ( ph /\ ps ) -> y R X )
23		bnj1152 29381	. . . . . . 7 |- ( y e. pred ( X , A , R ) <-> ( y e. A /\ y R X ) )
24	20, 22, 23	sylanbrc 662	. . . . . 6 |- ( ( ph /\ ps ) -> y e. pred ( X , A , R ) )
25	24, 19	elind 3627	. . . . 5 |- ( ( ph /\ ps ) -> y e. ( pred ( X , A , R ) i^i B ) )
26	15, 25	sseldd 3443	. . . 4 |- ( ( ph /\ ps ) -> y e. ( trCl ( X , A , R ) i^i B ) )
27		ne0i 3744	. . . 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 2688	. . 3 |- ( C =/= (/) <-> ( trCl ( X , A , R ) i^i B ) =/= (/) )
30	28, 29	sylibr 212	. 2 |- ( ( ph /\ ps ) -> C =/= (/) )
31		bnj893 29313	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) e. _V )
32	1, 11, 31	syl2anc 659	. . 3 |- ( ( ph /\ ps ) -> trCl ( X , A , R ) e. _V )
33		inex1g 4537	. . . 4 |- ( trCl ( X , A , R ) e. _V -> ( trCl ( X , A , R ) i^i B ) e. _V )
34	5, 33	syl5eqel 2494	. . 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 29161	1 |- ( ( ph /\ ps ) -> ( R Fr A /\ C C_ A /\ C =/= (/) /\ C e. _V ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   _Vcvv 3059   i^i cin 3413   C_ wss 3414   (/)c0 3738   class class class wbr 4395   Fr wfr 4779   /\ w-bnj17 29065   predc-bnj14 29067   Se w-bnj13 29069   FrSe w-bnj15 29071   trClc-bnj18 29073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-reg 8052  ax-inf2 8091

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-om 6684  df-1o 7167  df-bnj17 29066  df-bnj14 29068  df-bnj13 29070  df-bnj15 29072  df-bnj18 29074

This theorem is referenced by:  bnj1190  29391