Theorem nss 3523

Description: Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

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

Distinct variable groups:   x, A   x, B

Proof of Theorem nss

Step	Hyp	Ref	Expression
1		exanali 1638	. . 3 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
2		dfss2 3454	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
3	1, 2	xchbinxr 311	. 2 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A C_ B )
4	3	bicomi 202	1 |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368   E.wex 1587   e. wcel 1758   C_ wss 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3444  df-ss 3451

This theorem is referenced by:  grur1  9099  psslinpr  9312  reclem2pr  9329  mreexexlem2d  14703  prmcyg  16492  filcon  19589  alexsubALTlem4  19755  wilthlem2  22541  shne0i  25004  erdszelem10  27233  fundmpss  27722