Theorem encv 7028

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 7026	. 2 |- Rel ~~
2		brrelex12 4813	. 2 |- ( ( Rel ~~ /\ A ~~ B ) -> ( A e. _V /\ B e. _V ) )
3	1, 2	mpan 428	1 |- ( A ~~ B -> ( A e. _V /\ B e. _V ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   Rel wrel 4779   ~~ cen 7020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-en 7023

This theorem is used by:  bren  7030  en1uniel  7091  cardcl  7526  isnumi  7527  cardval3ex  7530  djuen  7567  ccfunen  7630