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Theorem exse 4843
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 4597 . . 3  |-  ( A  e.  V  ->  { y  e.  A  |  y R x }  e.  _V )
21ralrimivw 2879 . 2  |-  ( A  e.  V  ->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
3 df-se 4839 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
42, 3sylibr 212 1  |-  ( A  e.  V  ->  R Se  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113   class class class wbr 4447   Se wse 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-se 4839
This theorem is referenced by:  wemoiso  6770  wemoiso2  6771  oiiso  7963  hartogslem1  7968  oemapwe  8114  cantnffval2  8115  oemapweOLD  8136  cantnffval2OLD  8137  om2uzoi  12035  uzsinds  29149  bpolylem  29663
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