Theorem sscon 2168

Description: Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22.

Assertion

Ref	Expression
sscon	|- (A (_ B -> (C \ B) (_ (C \ A))

Proof of Theorem sscon

Step	Hyp	Ref	Expression
1		ssel 2060	. . . . 5 |- (A (_ B -> (x e. A -> x e. B))
2	1	con3d 95	. . . 4 |- (A (_ B -> (-. x e. B -> -. x e. A))
3	2	anim2d 560	. . 3 |- (A (_ B -> ((x e. C /\ -. x e. B) -> (x e. C /\ -. x e. A)))
4		eldif 2054	. . 3 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
5		eldif 2054	. . 3 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
6	3, 4, 5	3imtr4g 552	. 2 |- (A (_ B -> (x e. (C \ B) -> x e. (C \ A)))
7	6	ssrdv 2067	1 |- (A (_ B -> (C \ B) (_ (C \ A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 957   \ cdif 2041   (_ wss 2044

This theorem is referenced by:  sbthlem1 4436  sbthlem2 4437  fctop 7610  cctop 7612  clsval2 7645  ntrss 7648

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050

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