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Theorem frc 4540
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 4536 . . . 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 3641 . . 3  |-  ( { y  e.  B  | 
y R x }  =  (/)  <->  A. y  e.  B  -.  y R x )
65rexbii 2722 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   (/)c0 3620   class class class wbr 4204    Fr wfr 4530
This theorem is referenced by:  frirr  4551  epfrc  4560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-fr 4533
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