Theorem bnj1175 31072

Description: Technical lemma for bnj69 31078. 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
bnj1175.3	|- C = ( trCl ( X , A , R ) i^i B )
bnj1175.4	|- ( ch <-> ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ ( w e. A /\ w R z ) ) )
bnj1175.5	|- ( th <-> ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A ) )

Assertion

Ref	Expression
bnj1175	|- ( th -> ( w R z -> w e. trCl ( X , A , R ) ) )

Proof of Theorem bnj1175

Step	Hyp	Ref	Expression
1		bnj1175.4	. . . . 5 |- ( ch <-> ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ ( w e. A /\ w R z ) ) )
2		bnj255 30771	. . . . 5 |- ( ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A /\ w R z ) <-> ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ ( w e. A /\ w R z ) ) )
3		df-bnj17 30753	. . . . 5 |- ( ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A /\ w R z ) <-> ( ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A ) /\ w R z ) )
4	1, 2, 3	3bitr2i 288	. . . 4 |- ( ch <-> ( ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A ) /\ w R z ) )
5		bnj1175.5	. . . . 5 |- ( th <-> ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A ) )
6	5	anbi1i 731	. . . 4 |- ( ( th /\ w R z ) <-> ( ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) /\ ( R FrSe A /\ z e. A ) /\ w e. A ) /\ w R z ) )
7	4, 6	bitr4i 267	. . 3 |- ( ch <-> ( th /\ w R z ) )
8		bnj1125 31060	. . . . 5 |- ( ( R FrSe A /\ X e. A /\ z e. trCl ( X , A , R ) ) -> trCl ( z , A , R ) C_ trCl ( X , A , R ) )
9	1, 8	bnj835 30829	. . . 4 |- ( ch -> trCl ( z , A , R ) C_ trCl ( X , A , R ) )
10		bnj906 31000	. . . . . 6 |- ( ( R FrSe A /\ z e. A ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
11	1, 10	bnj836 30830	. . . . 5 |- ( ch -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
12		bnj1152 31066	. . . . . . 7 |- ( w e. pred ( z , A , R ) <-> ( w e. A /\ w R z ) )
13	12	biimpri 218	. . . . . 6 |- ( ( w e. A /\ w R z ) -> w e. pred ( z , A , R ) )
14	1, 13	bnj837 30831	. . . . 5 |- ( ch -> w e. pred ( z , A , R ) )
15	11, 14	sseldd 3604	. . . 4 |- ( ch -> w e. trCl ( z , A , R ) )
16	9, 15	sseldd 3604	. . 3 |- ( ch -> w e. trCl ( X , A , R ) )
17	7, 16	sylbir 225	. 2 |- ( ( th /\ w R z ) -> w e. trCl ( X , A , R ) )
18	17	ex 450	1 |- ( th -> ( w R z -> w e. trCl ( X , A , R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1190  31076