Theorem zfregfr 8509

Description: The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)

Assertion

Ref	Expression
zfregfr	|- _E Fr A

Proof of Theorem zfregfr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfepfr 5099	. 2 |- ( _E Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i y ) = (/) ) )
2		vex 3203	. . . . 5 |- x e. _V
3		zfreg 8500	. . . . 5 |- ( ( x e. _V /\ x =/= (/) ) -> E. y e. x ( y i^i x ) = (/) )
4	2, 3	mpan 706	. . . 4 |- ( x =/= (/) -> E. y e. x ( y i^i x ) = (/) )
5		incom 3805	. . . . . 6 |- ( y i^i x ) = ( x i^i y )
6	5	eqeq1i 2627	. . . . 5 |- ( ( y i^i x ) = (/) <-> ( x i^i y ) = (/) )
7	6	rexbii 3041	. . . 4 |- ( E. y e. x ( y i^i x ) = (/) <-> E. y e. x ( x i^i y ) = (/) )
8	4, 7	sylib 208	. . 3 |- ( x =/= (/) -> E. y e. x ( x i^i y ) = (/) )
9	8	adantl 482	. 2 |- ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i y ) = (/) )
10	1, 9	mpgbir 1726	1 |- _E Fr A

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   _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  ax-reg 8497

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:  en2lp  8510  dford2  8517  noinfep  8557  zfregs  8608  bnj852  30991  dford5reg  31687  trelpss  38659