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

Theorem zfregfr 8117
Description: The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
Assertion
Ref Expression
zfregfr  |-  _E  Fr  A

Proof of Theorem zfregfr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfepfr 4839 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y
)  =  (/) ) )
2 vex 3090 . . . 4  |-  x  e. 
_V
32zfreg2 8111 . . 3  |-  ( x  =/=  (/)  ->  E. y  e.  x  ( x  i^i  y )  =  (/) )
43adantl 467 . 2  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y )  =  (/) )
51, 4mpgbir 1669 1  |-  _E  Fr  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    =/= wne 2625   E.wrex 2783    i^i cin 3441    C_ wss 3442   (/)c0 3767    _E cep 4763    Fr wfr 4810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-reg 8107
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-eprel 4765  df-fr 4813
This theorem is referenced by:  en2lp  8118  dford2  8125  noinfep  8164  zfregs  8215  bnj852  29520  dford5reg  30215  trelpss  36444
  Copyright terms: Public domain W3C validator