Theorem sscon 3271

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 3151	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	con3d 631	. . . 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 3140	. . 3 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
5		eldif 3140	. . 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 3163	1 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2148   \ cdif 3128   C_ wss 3131

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144

This theorem is referenced by:  sscond  3274  sbthlem1  6958  sbthlem2  6959