Theorem bnj1190 29378

Description: Technical lemma for bnj69 29380. 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 1013	. . . . . 6 |- ( ph -> B C_ A )
3	2	adantr 463	. . . . 5 |- ( ( ph /\ ps ) -> B C_ A )
4		eqid 2402	. . . . . 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 1012	. . . . . . . . . 10 |- ( ph -> R FrSe A )
7	6	adantr 463	. . . . . . . . 9 |- ( ( ph /\ ps ) -> R FrSe A )
8	5	simp1bi 1012	. . . . . . . . . 10 |- ( ps -> x e. B )
9		ssel2 3436	. . . . . . . . . 10 |- ( ( B C_ A /\ x e. B ) -> x e. A )
10	2, 8, 9	syl2an 475	. . . . . . . . 9 |- ( ( ph /\ ps ) -> x e. A )
11	5, 4, 7, 3, 10	bnj1177 29376	. . . . . . . 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 29369	. . . . . . . 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 29375	. . . . . . 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 29374	. . . . . . 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 29373	. . . . . 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 29372	. . . . . 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 29371	. . . . 5 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. A -> ( v R u -> -. v e. B ) ) ) )
20	3, 19	bnj1171 29370	. . . 4 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. B -> -. v R u ) ) )
21	20	bnj1186 29377	. . 3 |- ( ( ph /\ ps ) -> E. u e. B A. v e. B -. v R u )
22	21	bnj1185 29166	. 2 |- ( ( ph /\ ps ) -> E. x e. B A. y e. B -. y R x )
23	22	bnj1185 29166	1 |- ( ( ph /\ ps ) -> E. w e. B A. z e. B -. z R w )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   e. wcel 1842   =/= wne 2598   A.wral 2753   E.wrex 2754   i^i cin 3412   C_ wss 3413   (/)c0 3737   class class class wbr 4394   FrSe w-bnj15 29058   trClc-bnj18 29060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-reg 8051  ax-inf2 8090

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-1o 7166  df-bnj17 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059  df-bnj18 29061  df-bnj19 29063

This theorem is referenced by:  bnj1189  29379