Theorem prcssprc 4230

Description: The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)

Assertion

Ref	Expression
prcssprc	|- ( ( A C_ B /\ A e/ _V ) -> B e/ _V )

Proof of Theorem prcssprc

Step	Hyp	Ref	Expression
1		ssexg 4228	. . . 4 |- ( ( A C_ B /\ B e. _V ) -> A e. _V )
2	1	ex 115	. . 3 |- ( A C_ B -> ( B e. _V -> A e. _V ) )
3	2	nelcon3d 2508	. 2 |- ( A C_ B -> ( A e/ _V -> B e/ _V ) )
4	3	imp 124	1 |- ( ( A C_ B /\ A e/ _V ) -> B e/ _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   e/ wnel 2497   _Vcvv 2802   C_ wss 3200

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-in1 619  ax-in2 620  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-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-nel 2498  df-v 2804  df-in 3206  df-ss 3213

This theorem is referenced by:  usgrprc  16106