Theorem frc 5080

Description: Property of well-founded relation (one direction of definition using class variables). (Contributed by NM, 17-Feb-2004.) (Revised by Mario Carneiro, 19-Nov-2014.)

Hypothesis

Ref	Expression
frc.1	|- B e. _V

Assertion

Ref	Expression
frc	|- ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B { y e. B | y R x } = (/) )

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

Proof of Theorem frc

Step	Hyp	Ref	Expression
1		frc.1	. . . 4 |- B e. _V
2		fri 5076	. . . 4 |- ( ( ( B e. _V /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )
3	1, 2	mpanl1 716	. . 3 |- ( ( R Fr A /\ ( B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )
4	3	3impb 1260	. 2 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B A. y e. B -. y R x )
5		rabeq0 3957	. . 3 |- ( { y e. B | y R x } = (/) <-> A. y e. B -. y R x )
6	5	rexbii 3041	. 2 |- ( E. x e. B { y e. B | y R x } = (/) <-> E. x e. B A. y e. B -. y R x )
7	4, 6	sylibr 224	1 |- ( ( R Fr A /\ B C_ A /\ B =/= (/) ) -> E. x e. B { y e. B | y R x } = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   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

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-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-fr 5073

This theorem is referenced by:  frirr  5091  epfrc  5100