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Theorem encv 6592
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 6590 . 2  |-  Rel  ~~
2 brrelex12 4535 . 2  |-  ( ( Rel  ~~  /\  A  ~~  B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
31, 2mpan 418 1  |-  ( A 
~~  B  ->  ( A  e.  _V  /\  B  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1461   _Vcvv 2655   class class class wbr 3893   Rel wrel 4502    ~~ cen 6584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-en 6587
This theorem is referenced by:  bren  6593  en1uniel  6650  cardcl  6984  isnumi  6985  cardval3ex  6988  djuen  7012
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