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Theorem exse 4139
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 3959 . . 3  |-  ( A  e.  V  ->  { y  e.  A  |  y R x }  e.  _V )
21ralrimivw 2443 . 2  |-  ( A  e.  V  ->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
3 df-se 4136 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
42, 3sylibr 132 1  |-  ( A  e.  V  ->  R Se  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1436   A.wral 2355   {crab 2359   _Vcvv 2615   class class class wbr 3822   Se wse 4132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-v 2617  df-in 2994  df-ss 3001  df-se 4136
This theorem is referenced by: (None)
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