Theorem sscon 3293

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 3173	. . . . 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 3162	. . 3 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
5		eldif 3162	. . 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 3185	1 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2164   \ cdif 3150   C_ wss 3153

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166

This theorem is referenced by:  sscond  3296  sbthlem1  7016  sbthlem2  7017