Theorem sscon 2171

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 2063	. . . . 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 561	. . 3 |- (A (_ B -> ((x e. C /\ -. x e. B) -> (x e. C /\ -. x e. A)))
4		eldif 2057	. . 3 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
5		eldif 2057	. . 3 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
6	3, 4, 5	3imtr4g 553	. 2 |- (A (_ B -> (x e. (C \ B) -> x e. (C \ A)))
7	6	ssrdv 2070	1 |- (A (_ B -> (C \ B) (_ (C \ A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 958   \ cdif 2044   (_ wss 2047

This theorem is referenced by:  sbthlem1 4447  sbthlem2 4448  fctopOLD 7650  cctop 7652  clsval2 7685  ntrss 7688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053

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