MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frc Unicode version

Theorem frc 4359
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 4355 . . . 4  |-  ( ( ( B  e.  _V  /\  R  Fr  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
31, 2mpanl1 661 . . 3  |-  ( ( R  Fr  A  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
433impb 1147 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
5 rabeq0 3476 . . 3  |-  ( { y  e.  B  | 
y R x }  =  (/)  <->  A. y  e.  B  -.  y R x )
65rexbii 2568 . 2  |-  ( E. x  e.  B  {
y  e.  B  | 
y R x }  =  (/)  <->  E. x  e.  B  A. y  e.  B  -.  y R x )
74, 6sylibr 203 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023    Fr wfr 4349
This theorem is referenced by:  frirr  4370  epfrc  4379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-fr 4352
  Copyright terms: Public domain W3C validator