Theorem sscon 3311

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 3191	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	con3d 632	. . . 4 |- ( A C_ B -> ( -. x e. B -> -. x e. A ) )
3	2	anim2d 337	. . 3 |- ( A C_ B -> ( ( x e. C /\ -. x e. B ) -> ( x e. C /\ -. x e. A ) ) )
4		eldif 3179	. . 3 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
5		eldif 3179	. . 3 |- ( x e. ( C \ A ) <-> ( x e. C /\ -. x e. A ) )
6	3, 4, 5	3imtr4g 205	. 2 |- ( A C_ B -> ( x e. ( C \ B ) -> x e. ( C \ A ) ) )
7	6	ssrdv 3203	1 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2177   \ cdif 3167   C_ wss 3170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183

This theorem is referenced by:  sscond  3314  sbthlem1  7074  sbthlem2  7075