Theorem nss 3522

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 1715	. . 3 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
2		dfss2 3453	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
3	1, 2	xchbinxr 312	. 2 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A C_ B )
4	3	bicomi 205	1 |- ( -. A C_ B <-> E. x ( x e. A /\ -. x e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435   E.wex 1659   e. wcel 1868   C_ wss 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-in 3443  df-ss 3450

This theorem is referenced by:  grur1  9245  psslinpr  9456  reclem2pr  9473  mreexexlem2d  15538  prmcyg  17515  filcon  20884  alexsubALTlem4  21051  wilthlem2  23980  shne0i  27086  erdszelem10  29918  fundmpss  30401