Theorem nss 3562

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 1647	. . 3 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
2		dfss2 3493	. . 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 1377   E.wex 1596   e. wcel 1767   C_ wss 3476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490

This theorem is referenced by:  grur1  9194  psslinpr  9405  reclem2pr  9422  mreexexlem2d  14893  prmcyg  16684  filcon  20116  alexsubALTlem4  20282  wilthlem2  23068  shne0i  26039  erdszelem10  28281  fundmpss  28770