Theorem bnj1190 32332

Description: Technical lemma for bnj69 32334. 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 1004	. . . . . 6 |- ( ph -> B C_ A )
3	2	adantr 465	. . . . 5 |- ( ( ph /\ ps ) -> B C_ A )
4		eqid 2454	. . . . . 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 1003	. . . . . . . . . 10 |- ( ph -> R FrSe A )
7	6	adantr 465	. . . . . . . . 9 |- ( ( ph /\ ps ) -> R FrSe A )
8	5	simp1bi 1003	. . . . . . . . . 10 |- ( ps -> x e. B )
9		ssel2 3460	. . . . . . . . . 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 32330	. . . . . . . 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 32323	. . . . . . . 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 32329	. . . . . . 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 32328	. . . . . . 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 32327	. . . . . 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 32326	. . . . . 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 32325	. . . . 5 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. A -> ( v R u -> -. v e. B ) ) ) )
20	3, 19	bnj1171 32324	. . . 4 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. B -> -. v R u ) ) )
21	20	bnj1186 32331	. . 3 |- ( ( ph /\ ps ) -> E. u e. B A. v e. B -. v R u )
22	21	bnj1185 32120	. 2 |- ( ( ph /\ ps ) -> E. x e. B A. y e. B -. y R x )
23	22	bnj1185 32120	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 965   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   i^i cin 3436   C_ wss 3437   (/)c0 3746   class class class wbr 4401   FrSe w-bnj15 32013   trClc-bnj18 32015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-reg 7919  ax-inf2 7959

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-1o 7031  df-bnj17 32008  df-bnj14 32010  df-bnj13 32012  df-bnj15 32014  df-bnj18 32016  df-bnj19 32018

This theorem is referenced by:  bnj1189  32333