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Theorem exse 4357
Description: Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
exse  |-  ( A  e.  V  ->  R Se  A )

Proof of Theorem exse
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rabexg 4164 . . 3  |-  ( A  e.  V  ->  { y  e.  A  |  y R x }  e.  _V )
21ralrimivw 2627 . 2  |-  ( A  e.  V  ->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
3 df-se 4353 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
42, 3sylibr 203 1  |-  ( A  e.  V  ->  R Se  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023   Se wse 4350
This theorem is referenced by:  wemoiso  5855  wemoiso2  5856  oiiso  7252  hartogslem1  7257  oemapwe  7396  cantnffval2  7397  om2uzoi  11018  uzsinds  24216  bpolylem  24783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-se 4353
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