Theorem ssexnelpss 3720

Description: If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020.)

Assertion

Ref	Expression
ssexnelpss	|- ( ( A C_ B /\ E. x e. B x e/ A ) -> A C. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssexnelpss

Step	Hyp	Ref	Expression
1		df-nel 2898	. . . 4 |- ( x e/ A <-> -. x e. A )
2		ssnelpss 3718	. . . . 5 |- ( A C_ B -> ( ( x e. B /\ -. x e. A ) -> A C. B ) )
3	2	expdimp 453	. . . 4 |- ( ( A C_ B /\ x e. B ) -> ( -. x e. A -> A C. B ) )
4	1, 3	syl5bi 232	. . 3 |- ( ( A C_ B /\ x e. B ) -> ( x e/ A -> A C. B ) )
5	4	rexlimdva 3031	. 2 |- ( A C_ B -> ( E. x e. B x e/ A -> A C. B ) )
6	5	imp 445	1 |- ( ( A C_ B /\ E. x e. B x e/ A ) -> A C. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   e/ wnel 2897   E.wrex 2913   C_ wss 3574   C. wpss 3575

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-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-pss 3590

This theorem is referenced by:  sgrpssmgm  17420  mndsssgrp  17421