Theorem bnj1177 29808

Description: Technical lemma for bnj69 29812. 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 29491	. . . 4 |- ( R FrSe A <-> ( R Fr A /\ R Se A ) )
3	2	simplbi 462	. . 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 29796	. . . . 5 |- trCl ( X , A , R ) C_ A
7		ssinss1 3659	. . . . 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 3461	. . 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 29734	. . . . . . 7 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
13	1, 11, 12	syl2anc 666	. . . . . 6 |- ( ( ph /\ ps ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
14		ssrin 3656	. . . . . 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 1023	. . . . . . . . 9 |- ( ps -> y e. B )
19	18	adantl 468	. . . . . . . 8 |- ( ( ph /\ ps ) -> y e. B )
20	16, 19	sseldd 3432	. . . . . . 7 |- ( ( ph /\ ps ) -> y e. A )
21	17	simp3bi 1024	. . . . . . . 8 |- ( ps -> y R X )
22	21	adantl 468	. . . . . . 7 |- ( ( ph /\ ps ) -> y R X )
23		bnj1152 29800	. . . . . . 7 |- ( y e. pred ( X , A , R ) <-> ( y e. A /\ y R X ) )
24	20, 22, 23	sylanbrc 669	. . . . . 6 |- ( ( ph /\ ps ) -> y e. pred ( X , A , R ) )
25	24, 19	elind 3617	. . . . 5 |- ( ( ph /\ ps ) -> y e. ( pred ( X , A , R ) i^i B ) )
26	15, 25	sseldd 3432	. . . 4 |- ( ( ph /\ ps ) -> y e. ( trCl ( X , A , R ) i^i B ) )
27		ne0i 3736	. . . 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 2687	. . 3 |- ( C =/= (/) <-> ( trCl ( X , A , R ) i^i B ) =/= (/) )
30	28, 29	sylibr 216	. 2 |- ( ( ph /\ ps ) -> C =/= (/) )
31		bnj893 29732	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) e. _V )
32	1, 11, 31	syl2anc 666	. . 3 |- ( ( ph /\ ps ) -> trCl ( X , A , R ) e. _V )
33		inex1g 4545	. . . 4 |- ( trCl ( X , A , R ) e. _V -> ( trCl ( X , A , R ) i^i B ) e. _V )
34	5, 33	syl5eqel 2532	. . 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 29580	1 |- ( ( ph /\ ps ) -> ( R Fr A /\ C C_ A /\ C =/= (/) /\ C e. _V ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   =/= wne 2621   _Vcvv 3044   i^i cin 3402   C_ wss 3403   (/)c0 3730   class class class wbr 4401   Fr wfr 4789   /\ w-bnj17 29484   predc-bnj14 29486   Se w-bnj13 29488   FrSe w-bnj15 29490   trClc-bnj18 29492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-reg 8104  ax-inf2 8143

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-1o 7179  df-bnj17 29485  df-bnj14 29487  df-bnj13 29489  df-bnj15 29491  df-bnj18 29493

This theorem is referenced by:  bnj1190  29810