Theorem bnj1190 33018

Description: Technical lemma for bnj69 33020. 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 1007	. . . . . 6 |- ( ph -> B C_ A )
3	2	adantr 465	. . . . 5 |- ( ( ph /\ ps ) -> B C_ A )
4		eqid 2460	. . . . . 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 1006	. . . . . . . . . 10 |- ( ph -> R FrSe A )
7	6	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ps ) -> R FrSe A )
8	5	simp1bi 1006	. . . . . . . . . 10 |- ( ps -> x e. B )
9		ssel2 3492	. . . . . . . . . 10 |- ( ( B C_ A /\ x e. B ) -> x e. A )
10	2, 8, 9	syl2an 477	. . . . . . . . 9 |- ( ( ph /\ ps ) -> x e. A )
11	5, 4, 7, 3, 10	bnj1177 33016	. . . . . . . 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 33009	. . . . . . . 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 33015	. . . . . . 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 236	. . . . . . . 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 236	. . . . . . . 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 33014	. . . . . . 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 33013	. . . . . 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 33012	. . . . . 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 33011	. . . . 5 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. A -> ( v R u -> -. v e. B ) ) ) )
20	3, 19	bnj1171 33010	. . . 4 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. B -> -. v R u ) ) )
21	20	bnj1186 33017	. . 3 |- ( ( ph /\ ps ) -> E. u e. B A. v e. B -. v R u )
22	21	bnj1185 32806	. 2 |- ( ( ph /\ ps ) -> E. x e. B A. y e. B -. y R x )
23	22	bnj1185 32806	1 |- ( ( ph /\ ps ) -> E. w e. B A. z e. B -. z R w )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   e. wcel 1762   =/= wne 2655   A.wral 2807   E.wrex 2808   i^i cin 3468   C_ wss 3469   (/)c0 3778   class class class wbr 4440   FrSe w-bnj15 32699   trClc-bnj18 32701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007  ax-inf2 8047

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-bnj17 32694  df-bnj14 32696  df-bnj13 32698  df-bnj15 32700  df-bnj18 32702  df-bnj19 32704

This theorem is referenced by:  bnj1189  33019