Theorem bnj1279 31086

Description: Technical lemma for bnj60 31130. 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
bnj1279.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1279.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1279.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1279.4	|- D = ( dom g i^i dom h )
bnj1279.5	|- E = { x e. D | ( g ` x ) =/= ( h ` x ) }
bnj1279.6	|- ( ph <-> ( R FrSe A /\ g e. C /\ h e. C /\ ( g |` D ) =/= ( h |` D ) ) )
bnj1279.7	|- ( ps <-> ( ph /\ x e. E /\ A. y e. E -. y R x ) )

Assertion

Ref	Expression
bnj1279	|- ( ( x e. E /\ A. y e. E -. y R x ) -> ( pred ( x , A , R ) i^i E ) = (/) )

Distinct variable groups:   y, A   y, E   y, R   x, y

Allowed substitution hints:   ph( x, y, f, g, h, d)   ps( x, y, f, g, h, d)   A( x, f, g, h, d)   B( x, y, f, g, h, d)   C( x, y, f, g, h, d)   D( x, y, f, g, h, d)   R( x, f, g, h, d)   E( x, f, g, h, d)   G( x, y, f, g, h, d)   Y( x, y, f, g, h, d)

Proof of Theorem bnj1279

Step	Hyp	Ref	Expression
1		n0 3931	. . . . . . . 8 |- ( ( pred ( x , A , R ) i^i E ) =/= (/) <-> E. y y e. ( pred ( x , A , R ) i^i E ) )
2		elin 3796	. . . . . . . . 9 |- ( y e. ( pred ( x , A , R ) i^i E ) <-> ( y e. pred ( x , A , R ) /\ y e. E ) )
3	2	exbii 1774	. . . . . . . 8 |- ( E. y y e. ( pred ( x , A , R ) i^i E ) <-> E. y ( y e. pred ( x , A , R ) /\ y e. E ) )
4	1, 3	sylbb 209	. . . . . . 7 |- ( ( pred ( x , A , R ) i^i E ) =/= (/) -> E. y ( y e. pred ( x , A , R ) /\ y e. E ) )
5		df-bnj14 30755	. . . . . . . . 9 |- pred ( x , A , R ) = { y e. A | y R x }
6	5	bnj1538 30925	. . . . . . . 8 |- ( y e. pred ( x , A , R ) -> y R x )
7	6	anim1i 592	. . . . . . 7 |- ( ( y e. pred ( x , A , R ) /\ y e. E ) -> ( y R x /\ y e. E ) )
8	4, 7	bnj593 30815	. . . . . 6 |- ( ( pred ( x , A , R ) i^i E ) =/= (/) -> E. y ( y R x /\ y e. E ) )
9	8	3ad2ant3 1084	. . . . 5 |- ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) -> E. y ( y R x /\ y e. E ) )
10		nfv 1843	. . . . . . 7 |- F/ y x e. E
11		nfra1 2941	. . . . . . 7 |- F/ y A. y e. E -. y R x
12		nfv 1843	. . . . . . 7 |- F/ y ( pred ( x , A , R ) i^i E ) =/= (/)
13	10, 11, 12	nf3an 1831	. . . . . 6 |- F/ y ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) )
14	13	nf5ri 2065	. . . . 5 |- ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) -> A. y ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) )
15	9, 14	bnj1275 30884	. . . 4 |- ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) -> E. y ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) /\ y R x /\ y e. E ) )
16		simp2 1062	. . . 4 |- ( ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) /\ y R x /\ y e. E ) -> y R x )
17		simp12 1092	. . . . 5 |- ( ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) /\ y R x /\ y e. E ) -> A. y e. E -. y R x )
18		simp3 1063	. . . . 5 |- ( ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) /\ y R x /\ y e. E ) -> y e. E )
19	17, 18	bnj1294 30888	. . . 4 |- ( ( ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) ) /\ y R x /\ y e. E ) -> -. y R x )
20	15, 16, 19	bnj1304 30890	. . 3 |- -. ( x e. E /\ A. y e. E -. y R x /\ ( pred ( x , A , R ) i^i E ) =/= (/) )
21	20	bnj1224 30872	. 2 |- ( ( x e. E /\ A. y e. E -. y R x ) -> -. ( pred ( x , A , R ) i^i E ) =/= (/) )
22		nne 2798	. 2 |- ( -. ( pred ( x , A , R ) i^i E ) =/= (/) <-> ( pred ( x , A , R ) i^i E ) = (/) )
23	21, 22	sylib 208	1 |- ( ( x e. E /\ A. y e. E -. y R x ) -> ( pred ( x , A , R ) i^i E ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-bnj14 30755

This theorem is referenced by:  bnj1311  31092