Theorem epfrc 5100

Description: A subset of an epsilon-founded class has a minimal element. (Contributed by NM, 17-Feb-2004.) (Revised by David Abernethy, 22-Feb-2011.)

Hypothesis

Ref	Expression
epfrc.1	|- B e. _V

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem epfrc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		epfrc.1	. . 3 |- B e. _V
2	1	frc 5080	. 2 |- ( ( _E Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B { y e. B | y _E x } = (/) )
3		dfin5 3582	. . . . 5 |- ( B i^i x ) = { y e. B | y e. x }
4		epel 5032	. . . . . . 7 |- ( y _E x <-> y e. x )
5	4	a1i 11	. . . . . 6 |- ( y e. B -> ( y _E x <-> y e. x ) )
6	5	rabbiia 3185	. . . . 5 |- { y e. B | y _E x } = { y e. B | y e. x }
7	3, 6	eqtr4i 2647	. . . 4 |- ( B i^i x ) = { y e. B | y _E x }
8	7	eqeq1i 2627	. . 3 |- ( ( B i^i x ) = (/) <-> { y e. B | y _E x } = (/) )
9	8	rexbii 3041	. 2 |- ( E. x e. B ( B i^i x ) = (/) <-> E. x e. B { y e. B | y _E x } = (/) )
10	2, 9	sylibr 224	1 |- ( ( _E Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B ( B i^i x ) = (/) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   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:  wefrc  5108  onfr  5763  epfrs  8607