Theorem encv 6914

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 6912	. 2 |- Rel ~~
2		brrelex12 4764	. 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 2202   _Vcvv 2802   class class class wbr 4088   Rel wrel 4730   ~~ cen 6906

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-en 6909

This theorem is referenced by:  bren  6916  en1uniel  6977  cardcl  7384  isnumi  7385  cardval3ex  7388  djuen  7425  ccfunen  7482