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Theorem encv 6461
Description: If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
Assertion
Ref Expression
encv  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )

Proof of Theorem encv
StepHypRef Expression
1 relen 6459 . 2  |-  Rel  ~~
2 brrelex12 4475 . 2  |-  ( ( Rel  ~~  /\  A  ~~  B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
31, 2mpan 415 1  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   _Vcvv 2619   class class class wbr 3845   Rel wrel 4443    ~~ cen 6453
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-en 6456
This theorem is referenced by:  bren  6462  en1uniel  6519  cardcl  6807  isnumi  6808  cardval3ex  6811
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