Theorem sscon 3241

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 3122	. . . . 5 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	con3d 621	. . . 4 |- ( A C_ B -> ( -. x e. B -> -. x e. A ) )
3	2	anim2d 335	. . 3 |- ( A C_ B -> ( ( x e. C /\ -. x e. B ) -> ( x e. C /\ -. x e. A ) ) )
4		eldif 3111	. . 3 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
5		eldif 3111	. . 3 |- ( x e. ( C \ A ) <-> ( x e. C /\ -. x e. A ) )
6	3, 4, 5	3imtr4g 204	. 2 |- ( A C_ B -> ( x e. ( C \ B ) -> x e. ( C \ A ) ) )
7	6	ssrdv 3134	1 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2128   \ cdif 3099   C_ wss 3102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115

This theorem is referenced by:  sscond  3244  sbthlem1  6894  sbthlem2  6895