Theorem nss 3557

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 1671	. . 3 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
2		dfss2 3488	. . 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 1393   E.wex 1613   e. wcel 1819   C_ wss 3471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-in 3478  df-ss 3485

This theorem is referenced by:  grur1  9215  psslinpr  9426  reclem2pr  9443  mreexexlem2d  15061  prmcyg  17022  filcon  20509  alexsubALTlem4  20675  wilthlem2  23468  shne0i  26492  erdszelem10  28819  fundmpss  29371