Theorem sscon 3401

Description: Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
sscon	|- (A C_ B -> (C \ B) C_ (C \ A))

Proof of Theorem sscon

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . 5 |- (A C_ B -> (x e. A -> x e. B))
2	1	con3d 125	. . . 4 |- (A C_ B -> (-. x e. B -> -. x e. A))
3	2	anim2d 548	. . 3 |- (A C_ B -> ((x e. C /\ -. x e. B) -> (x e. C /\ -. x e. A)))
4		eldif 3222	. . 3 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
5		eldif 3222	. . 3 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
6	3, 4, 5	3imtr4g 261	. 2 |- (A C_ B -> (x e. (C \ B) -> x e. (C \ A)))
7	6	ssrdv 3279	1 |- (A C_ B -> (C \ B) C_ (C \ A))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   e. wcel 1710   \ cdif 3207   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-dif 3216  df-ss 3260

This theorem is referenced by:  sscond  3404