Theorem nssd 39288

Description: Negation of subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
nssd.1	|- ( ph -> X e. A )
nssd.2	|- ( ph -> -. X e. B )

Assertion

Ref	Expression
nssd	|- ( ph -> -. A C_ B )

Proof of Theorem nssd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nssd.1	. . 3 |- ( ph -> X e. A )
2		nssd.2	. . . 4 |- ( ph -> -. X e. B )
3	1, 2	jca 554	. . 3 |- ( ph -> ( X e. A /\ -. X e. B ) )
4		eleq1 2689	. . . . 5 |- ( x = X -> ( x e. A <-> X e. A ) )
5		eleq1 2689	. . . . . 6 |- ( x = X -> ( x e. B <-> X e. B ) )
6	5	notbid 308	. . . . 5 |- ( x = X -> ( -. x e. B <-> -. X e. B ) )
7	4, 6	anbi12d 747	. . . 4 |- ( x = X -> ( ( x e. A /\ -. x e. B ) <-> ( X e. A /\ -. X e. B ) ) )
8	7	spcegv 3294	. . 3 |- ( X e. A -> ( ( X e. A /\ -. X e. B ) -> E. x ( x e. A /\ -. x e. B ) ) )
9	1, 3, 8	sylc 65	. 2 |- ( ph -> E. x ( x e. A /\ -. x e. B ) )
10		nss 3663	. 2 |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) )
11	9, 10	sylibr 224	1 |- ( ph -> -. A C_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   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

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-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by: (None)