Theorem difdif 1595

Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22.

Assertion

Ref	Expression
difdif	|- (A \ (B \ A)) = A

Proof of Theorem difdif

Step	Hyp	Ref	Expression
1		eldif 1496	. . . . . 6 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
2	1	negbii 162	. . . . 5 |- (-. x e. (B \ A) <-> -. (x e. B /\ -. x e. A))
3		iman 205	. . . . 5 |- ((x e. B -> x e. A) <-> -. (x e. B /\ -. x e. A))
4	2, 3	bitr4 154	. . . 4 |- (-. x e. (B \ A) <-> (x e. B -> x e. A))
5	4	anbi2i 367	. . 3 |- ((x e. A /\ -. x e. (B \ A)) <-> (x e. A /\ (x e. B -> x e. A)))
6		pm4.45im 267	. . 3 |- (x e. A <-> (x e. A /\ (x e. B -> x e. A)))
7	5, 6	bitr4 154	. 2 |- ((x e. A /\ -. x e. (B \ A)) <-> x e. A)
8	7	difeqri 1589	1 |- (A \ (B \ A)) = A

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092   \ cdif 1484

This theorem is referenced by:  dif0 1756

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489

metamath.org