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Theorem frc 4800
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 4796 . . . 4  |-  ( ( ( B  e.  _V  /\  R  Fr  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
31, 2mpanl1 686 . . 3  |-  ( ( R  Fr  A  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
433impb 1204 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
5 rabeq0 3754 . . 3  |-  ( { y  e.  B  | 
y R x }  =  (/)  <->  A. y  e.  B  -.  y R x )
65rexbii 2889 . 2  |-  ( E. x  e.  B  {
y  e.  B  | 
y R x }  =  (/)  <->  E. x  e.  B  A. y  e.  B  -.  y R x )
74, 6sylibr 216 1  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  { y  e.  B  |  y R x }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   (/)c0 3731   class class class wbr 4402    Fr wfr 4790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-fr 4793
This theorem is referenced by:  frirr  4811  epfrc  4820
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