Theorem ssdif0 3610

Description: Subclass expressed in terms of difference. Exercise 7 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Assertion

Ref	Expression
ssdif0	|- (A C_ B <-> (A \ B) = (/))

Proof of Theorem ssdif0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iman 413	. . . 4 |- ((x e. A -> x e. B) <-> -. (x e. A /\ -. x e. B))
2		eldif 3222	. . . 4 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
3	1, 2	xchbinxr 302	. . 3 |- ((x e. A -> x e. B) <-> -. x e. (A \ B))
4	3	albii 1566	. 2 |- (A.x(x e. A -> x e. B) <-> A.x -. x e. (A \ B))
5		dfss2 3263	. 2 |- (A C_ B <-> A.x(x e. A -> x e. B))
6		eq0 3565	. 2 |- ((A \ B) = (/) <-> A.x -. x e. (A \ B))
7	4, 5, 6	3bitr4i 268	1 |- (A C_ B <-> (A \ B) = (/))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   \ cdif 3207   C_ wss 3258  (/)c0 3551

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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  vdif0  3611  pssdifn0  3612  difid  3619  difin0  3624  sfinltfin  4536