Theorem prcssprc 4806

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 4804	. . . 4 |- ( ( A C_ B /\ B e. _V ) -> A e. _V )
2	1	ex 450	. . 3 |- ( A C_ B -> ( B e. _V -> A e. _V ) )
3	2	nelcon3d 2909	. 2 |- ( A C_ B -> ( A e/ _V -> B e/ _V ) )
4	3	imp 445	1 |- ( ( A C_ B /\ A e/ _V ) -> B e/ _V )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   e/ wnel 2897   _Vcvv 3200   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  usgrprc  26158  rgrusgrprc  26485  rgrprc  26487