Theorem encv 6983

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

Step	Hyp	Ref	Expression
1		relen 6981	. 2 |- Rel ~~
2		brrelex12 4790	. 2 |- ( ( Rel ~~ /\ A ~~ B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 424	1 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   _Vcvv 2815   class class class wbr 4111   Rel wrel 4756   ~~ cen 6975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-en 6978

This theorem is referenced by:  bren  6985  en1uniel  7046  cardcl  7479  isnumi  7480  cardval3ex  7483  djuen  7520  ccfunen  7580