Theorem bnj1190 31076

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
bnj1190.1	|- ( ph <-> ( R FrSe A /\ B C_ A /\ B =/= (/) ) )
bnj1190.2	|- ( ps <-> ( x e. B /\ y e. B /\ y R x ) )

Assertion

Ref	Expression
bnj1190	|- ( ( ph /\ ps ) -> E. w e. B A. z e. B -. z R w )

Distinct variable groups:   w, B, x, z   y, B, x, z   w, R, x, z   y, R

Allowed substitution hints:   ph( x, y, z, w)   ps( x, y, z, w)   A( x, y, z, w)

Proof of Theorem bnj1190

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1190.1	. . . . . . 7 |- ( ph <-> ( R FrSe A /\ B C_ A /\ B =/= (/) ) )
2	1	simp2bi 1077	. . . . . 6 |- ( ph -> B C_ A )
3	2	adantr 481	. . . . 5 |- ( ( ph /\ ps ) -> B C_ A )
4		eqid 2622	. . . . . 6 |- ( trCl ( x , A , R ) i^i B ) = ( trCl ( x , A , R ) i^i B )
5		bnj1190.2	. . . . . . . . 9 |- ( ps <-> ( x e. B /\ y e. B /\ y R x ) )
6	1	simp1bi 1076	. . . . . . . . . 10 |- ( ph -> R FrSe A )
7	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ps ) -> R FrSe A )
8	5	simp1bi 1076	. . . . . . . . . 10 |- ( ps -> x e. B )
9		ssel2 3598	. . . . . . . . . 10 |- ( ( B C_ A /\ x e. B ) -> x e. A )
10	2, 8, 9	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ ps ) -> x e. A )
11	5, 4, 7, 3, 10	bnj1177 31074	. . . . . . . 8 |- ( ( ph /\ ps ) -> ( R Fr A /\ ( trCl ( x , A , R ) i^i B ) C_ A /\ ( trCl ( x , A , R ) i^i B ) =/= (/) /\ ( trCl ( x , A , R ) i^i B ) e. _V ) )
12		bnj1154 31067	. . . . . . . 8 |- ( ( R Fr A /\ ( trCl ( x , A , R ) i^i B ) C_ A /\ ( trCl ( x , A , R ) i^i B ) =/= (/) /\ ( trCl ( x , A , R ) i^i B ) e. _V ) -> E. u e. ( trCl ( x , A , R ) i^i B ) A. v e. ( trCl ( x , A , R ) i^i B ) -. v R u )
13	11, 12	bnj1176 31073	. . . . . . 7 |- E. u A. v ( ( ph /\ ps ) -> ( u e. ( trCl ( x , A , R ) i^i B ) /\ ( ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ v e. A ) -> ( v R u -> -. v e. ( trCl ( x , A , R ) i^i B ) ) ) ) )
14		biid 251	. . . . . . . 8 |- ( ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ ( v e. A /\ v R u ) ) <-> ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ ( v e. A /\ v R u ) ) )
15		biid 251	. . . . . . . 8 |- ( ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ v e. A ) <-> ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ v e. A ) )
16	4, 14, 15	bnj1175 31072	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ v e. A ) -> ( v R u -> v e. trCl ( x , A , R ) ) )
17	4, 13, 16	bnj1174 31071	. . . . . 6 |- E. u A. v ( ( ph /\ ps ) -> ( ( ph /\ ps /\ u e. ( trCl ( x , A , R ) i^i B ) ) /\ ( ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ v e. A ) -> ( v R u -> -. v e. B ) ) ) )
18	4, 15, 7, 10	bnj1173 31070	. . . . . 6 |- ( ( ph /\ ps /\ u e. ( trCl ( x , A , R ) i^i B ) ) -> ( ( ( R FrSe A /\ x e. A /\ u e. trCl ( x , A , R ) ) /\ ( R FrSe A /\ u e. A ) /\ v e. A ) <-> v e. A ) )
19	4, 17, 18	bnj1172 31069	. . . . 5 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. A -> ( v R u -> -. v e. B ) ) ) )
20	3, 19	bnj1171 31068	. . . 4 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. B -> -. v R u ) ) )
21	20	bnj1186 31075	. . 3 |- ( ( ph /\ ps ) -> E. u e. B A. v e. B -. v R u )
22	21	bnj1185 30864	. 2 |- ( ( ph /\ ps ) -> E. x e. B A. y e. B -. y R x )
23	22	bnj1185 30864	1 |- ( ( ph /\ ps ) -> E. w e. B A. z e. B -. z R w )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   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:  bnj1189  31077