Theorem encv 6800

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 6798	. 2 |- Rel ~~
2		brrelex12 4697	. 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 2164   _Vcvv 2760   class class class wbr 4029   Rel wrel 4664   ~~ cen 6792

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-en 6795

This theorem is referenced by:  bren  6801  en1uniel  6858  cardcl  7241  isnumi  7242  cardval3ex  7245  djuen  7271  ccfunen  7324