Theorem dfepfr 5099

Description: An alternate way of saying that the epsilon relation is well-founded. (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 23-Jun-2015.)

Assertion

Ref	Expression
dfepfr	|- ( _E Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i y ) = (/) ) )

Distinct variable group:   x, y, A

Proof of Theorem dfepfr

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffr2 5079	. 2 |- ( _E Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x { z e. x | z _E y } = (/) ) )
2		epel 5032	. . . . . . . . 9 |- ( z _E y <-> z e. y )
3	2	a1i 11	. . . . . . . 8 |- ( z e. x -> ( z _E y <-> z e. y ) )
4	3	rabbiia 3185	. . . . . . 7 |- { z e. x | z _E y } = { z e. x | z e. y }
5		dfin5 3582	. . . . . . 7 |- ( x i^i y ) = { z e. x | z e. y }
6	4, 5	eqtr4i 2647	. . . . . 6 |- { z e. x | z _E y } = ( x i^i y )
7	6	eqeq1i 2627	. . . . 5 |- ( { z e. x | z _E y } = (/) <-> ( x i^i y ) = (/) )
8	7	rexbii 3041	. . . 4 |- ( E. y e. x { z e. x | z _E y } = (/) <-> E. y e. x ( x i^i y ) = (/) )
9	8	imbi2i 326	. . 3 |- ( ( ( x C_ A /\ x =/= (/) ) -> E. y e. x { z e. x | z _E y } = (/) ) <-> ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i y ) = (/) ) )
10	9	albii 1747	. 2 |- ( A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x { z e. x | z _E y } = (/) ) <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i y ) = (/) ) )
11	1, 10	bitri 264	1 |- ( _E Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i y ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   =/= wne 2794   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   _E cep 5028   Fr wfr 5070

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073

This theorem is referenced by:  onfr  5763  zfregfr  8509  onfrALTlem3  38759  onfrALT  38764  onfrALTlem3VD  39123  onfrALTVD  39127