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Theorem zfregfr 8046
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 4873 . 2  |-  (  _E  Fr  A  <->  A. x
( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y
)  =  (/) ) )
2 vex 3112 . . . 4  |-  x  e. 
_V
32zfreg2 8040 . . 3  |-  ( x  =/=  (/)  ->  E. y  e.  x  ( x  i^i  y )  =  (/) )
43adantl 466 . 2  |-  ( ( x  C_  A  /\  x  =/=  (/) )  ->  E. y  e.  x  ( x  i^i  y )  =  (/) )
51, 4mpgbir 1623 1  |-  _E  Fr  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    =/= wne 2652   E.wrex 2808    i^i cin 3470    C_ wss 3471   (/)c0 3793    _E cep 4798    Fr wfr 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-reg 8036
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-eprel 4800  df-fr 4847
This theorem is referenced by:  en2lp  8047  dford2  8054  noinfep  8093  noinfepOLD  8094  zfregs  8180  dford5reg  29431  trelpss  31568  bnj852  34122
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