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

Theorem frc 4805
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 4801 . . . 4  |-  ( ( ( B  e.  _V  /\  R  Fr  A )  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
31, 2mpanl1 694 . . 3  |-  ( ( R  Fr  A  /\  ( B  C_  A  /\  B  =/=  (/) ) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
433impb 1227 . 2  |-  ( ( R  Fr  A  /\  B  C_  A  /\  B  =/=  (/) )  ->  E. x  e.  B  A. y  e.  B  -.  y R x )
5 rabeq0 3757 . . 3  |-  ( { y  e.  B  | 
y R x }  =  (/)  <->  A. y  e.  B  -.  y R x )
65rexbii 2881 . 2  |-  ( E. x  e.  B  {
y  e.  B  | 
y R x }  =  (/)  <->  E. x  e.  B  A. y  e.  B  -.  y R x )
74, 6sylibr 217 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   class class class wbr 4395    Fr wfr 4795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-fr 4798
This theorem is referenced by:  frirr  4816  epfrc  4825
  Copyright terms: Public domain W3C validator