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Theorem exse 4073
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 3900 . . 3  |-  ( A  e.  V  ->  { y  e.  A  |  y R x }  e.  _V )
21ralrimivw 2393 . 2  |-  ( A  e.  V  ->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
3 df-se 4070 . 2  |-  ( R Se  A  <->  A. x  e.  A  { y  e.  A  |  y R x }  e.  _V )
42, 3sylibr 137 1  |-  ( A  e.  V  ->  R Se  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1393   A.wral 2306   {crab 2310   _Vcvv 2557   class class class wbr 3764   Se wse 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-se 4070
This theorem is referenced by: (None)
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