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Theorem frc 4491
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
StepHypRef Expression
1 frc.1 . . . 4  |-  B  e. 
_V
2 fri 4487 . . . 4  |-  ( ( ( B  e.  _V  /\  R  Fr  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
31, 2mpanl1 662 . . 3  |-  ( ( R  Fr  A  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
433impb 1149 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
5 rabeq0 3594 . . 3  |-  ( { y  e.  B  | 
y R x }  =  (/)  <->  A. y  e.  B  -.  y R x )
65rexbii 2676 . 2  |-  ( E. x  e.  B  {
y  e.  B  | 
y R x }  =  (/)  <->  E. x  e.  B  A. y  e.  B  -.  y R x )
74, 6sylibr 204 1  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y R x }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   E.wrex 2652   {crab 2655   _Vcvv 2901    C_ wss 3265   (/)c0 3573   class class class wbr 4155    Fr wfr 4481
This theorem is referenced by:  frirr  4502  epfrc  4511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-in 3272  df-ss 3279  df-nul 3574  df-fr 4484
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