Theorem difin 2245

Description: Difference with intersection. Theorem 33 of [Suppes] p. 29.

Assertion

Ref	Expression
difin	|- (A \ (A i^i B)) = (A \ B)

Proof of Theorem difin

Step	Hyp	Ref	Expression
1		abai 479	. . . 4 |- ((x e. A /\ -. x e. B) <-> (x e. A /\ (x e. A -> -. x e. B)))
2		imnan 242	. . . . 5 |- ((x e. A -> -. x e. B) <-> -. (x e. A /\ x e. B))
3	2	anbi2i 480	. . . 4 |- ((x e. A /\ (x e. A -> -. x e. B)) <-> (x e. A /\ -. (x e. A /\ x e. B)))
4	1, 3	bitr 173	. . 3 |- ((x e. A /\ -. x e. B) <-> (x e. A /\ -. (x e. A /\ x e. B)))
5		eldif 2057	. . 3 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
6		eldif 2057	. . . 4 |- (x e. (A \ (A i^i B)) <-> (x e. A /\ -. x e. (A i^i B)))
7		elin 2207	. . . . . 6 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
8	7	negbii 187	. . . . 5 |- (-. x e. (A i^i B) <-> -. (x e. A /\ x e. B))
9	8	anbi2i 480	. . . 4 |- ((x e. A /\ -. x e. (A i^i B)) <-> (x e. A /\ -. (x e. A /\ x e. B)))
10	6, 9	bitr 173	. . 3 |- (x e. (A \ (A i^i B)) <-> (x e. A /\ -. (x e. A /\ x e. B)))
11	4, 5, 10	3bitr4r 184	. 2 |- (x e. (A \ (A i^i B)) <-> x e. (A \ B))
12	11	eqriv 1474	1 |- (A \ (A i^i B)) = (A \ B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   \ cdif 2044   i^i cin 2046

This theorem is referenced by:  dfin4 2248  indif 2250  symdif1 2265  dfsdom2 4460

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

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