Theorem encv 6833

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 6831	. 2 |- Rel ~~
2		brrelex12 4713	. 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 2176   _Vcvv 2772   class class class wbr 4044   Rel wrel 4680   ~~ cen 6825

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-en 6828

This theorem is referenced by:  bren  6835  en1uniel  6896  cardcl  7288  isnumi  7289  cardval3ex  7292  djuen  7323  ccfunen  7376