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Theorem epsetlike 24935
Description: The epsilon relationship is set-like. (Contributed by Scott Fenton, 27-Mar-2011.)
Assertion
Ref Expression
epsetlike  |-  A. x  e.  A  Pred (  _E  ,  A ,  x
)  e.  _V

Proof of Theorem epsetlike
StepHypRef Expression
1 predep 24933 . . . 4  |-  ( x  e.  A  ->  Pred (  _E  ,  A ,  x
)  =  ( A  i^i  x ) )
2 incom 3449 . . . 4  |-  ( A  i^i  x )  =  ( x  i^i  A
)
31, 2syl6eq 2414 . . 3  |-  ( x  e.  A  ->  Pred (  _E  ,  A ,  x
)  =  ( x  i^i  A ) )
4 inex1g 4259 . . 3  |-  ( x  e.  A  ->  (
x  i^i  A )  e.  _V )
53, 4eqeltrd 2440 . 2  |-  ( x  e.  A  ->  Pred (  _E  ,  A ,  x
)  e.  _V )
65rgen 2693 1  |-  A. x  e.  A  Pred (  _E  ,  A ,  x
)  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1715   A.wral 2628   _Vcvv 2873    i^i cin 3237    _E cep 4406   Predcpred 24908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-eprel 4408  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-pred 24909
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