Theorem nss 3330

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 1585	. . 3 |- (E.x(x e. A /\ -. x e. B) <-> -. A.x(x e. A -> x e. B))
2		dfss2 3263	. . 3 |- (A C_ B <-> A.x(x e. A -> x e. B))
3	1, 2	xchbinxr 302	. 2 |- (E.x(x e. A /\ -. x e. B) <-> -. A C_ B)
4	3	bicomi 193	1 |- (-. A C_ B <-> E.x(x e. A /\ -. x e. B))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   e. wcel 1710   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)