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

Theorem frc 4850
 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
Assertion
Ref Expression
frc
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem frc
StepHypRef Expression
1 frc.1 . . . 4
2 fri 4846 . . . 4
31, 2mpanl1 680 . . 3
433impb 1192 . 2
5 rabeq0 3807 . . 3
65rexbii 2959 . 2
74, 6sylibr 212 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  {crab 2811   cvv 3109  C_wss 3475   c0 3784   class class class wbr 4452  Frwfr 4840 This theorem is referenced by:  frirr  4861  epfrc  4870 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-fr 4843
 Copyright terms: Public domain W3C validator