Theorem nss 3489

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 1720	. . 3 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
2		dfss2 3420	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
3	1, 2	xchbinxr 313	. 2 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A C_ B )
4	3	bicomi 206	1 |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1441   E.wex 1662   e. wcel 1886   C_ wss 3403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-in 3410  df-ss 3417

This theorem is referenced by:  grur1  9242  psslinpr  9453  reclem2pr  9470  mreexexlem2d  15544  prmcyg  17521  filcon  20891  alexsubALTlem4  21058  wilthlem2  23987  shne0i  27094  erdszelem10  29916  fundmpss  30400  relintabex  36181