Theorem bnj1190 29817

Description: Technical lemma for bnj69 29819. 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 1024	. . . . . 6 |- ( ph -> B C_ A )
3	2	adantr 467	. . . . 5 |- ( ( ph /\ ps ) -> B C_ A )
4		eqid 2451	. . . . . 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 1023	. . . . . . . . . 10 |- ( ph -> R FrSe A )
7	6	adantr 467	. . . . . . . . 9 |- ( ( ph /\ ps ) -> R FrSe A )
8	5	simp1bi 1023	. . . . . . . . . 10 |- ( ps -> x e. B )
9		ssel2 3427	. . . . . . . . . 10 |- ( ( B C_ A /\ x e. B ) -> x e. A )
10	2, 8, 9	syl2an 480	. . . . . . . . 9 |- ( ( ph /\ ps ) -> x e. A )
11	5, 4, 7, 3, 10	bnj1177 29815	. . . . . . . 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 29808	. . . . . . . 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 29814	. . . . . . 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 240	. . . . . . . 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 240	. . . . . . . 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 29813	. . . . . . 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 29812	. . . . . 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 29811	. . . . . 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 29810	. . . . 5 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. A -> ( v R u -> -. v e. B ) ) ) )
20	3, 19	bnj1171 29809	. . . 4 |- E. u A. v ( ( ph /\ ps ) -> ( u e. B /\ ( v e. B -> -. v R u ) ) )
21	20	bnj1186 29816	. . 3 |- ( ( ph /\ ps ) -> E. u e. B A. v e. B -. v R u )
22	21	bnj1185 29605	. 2 |- ( ( ph /\ ps ) -> E. x e. B A. y e. B -. y R x )
23	22	bnj1185 29605	1 |- ( ( ph /\ ps ) -> E. w e. B A. z e. B -. z R w )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   i^i cin 3403   C_ wss 3404   (/)c0 3731   class class class wbr 4402   FrSe w-bnj15 29497   trClc-bnj18 29499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-bnj17 29492  df-bnj14 29494  df-bnj13 29496  df-bnj15 29498  df-bnj18 29500  df-bnj19 29502

This theorem is referenced by:  bnj1189  29818