Theorem sscon 3306

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 3186	. . . . 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 3174	. . 3 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
5		eldif 3174	. . 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 3198	1 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2175   \ cdif 3162   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178

This theorem is referenced by:  sscond  3309  sbthlem1  7058  sbthlem2  7059