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Theorem exse 4373
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 4180 . . 3  |-  ( A  e.  V  ->  { y  e.  A  |  y R x }  e.  _V )
21ralrimivw 2640 . 2  |-  ( A  e.  V  ->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
3 df-se 4369 . 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 1696   A.wral 2556   {crab 2560   _Vcvv 2801   class class class wbr 4039   Se wse 4366
This theorem is referenced by:  wemoiso  5871  wemoiso2  5872  oiiso  7268  hartogslem1  7273  oemapwe  7412  cantnffval2  7413  om2uzoi  11034  uzsinds  24287  bpolylem  24855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-se 4369
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