Theorem difdif 3354

Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Assertion

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

Proof of Theorem difdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( x e. A /\ -. x e. ( B \ A ) ) -> x e. A )
2		pm4.45im 334	. . . 4 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
3		imanim 699	. . . . . 6 |- ( ( x e. B -> x e. A ) -> -. ( x e. B /\ -. x e. A ) )
4		eldif 3229	. . . . . 6 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
5	3, 4	sylnibr 688	. . . . 5 |- ( ( x e. B -> x e. A ) -> -. x e. ( B \ A ) )
6	5	anim2i 342	. . . 4 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) -> ( x e. A /\ -. x e. ( B \ A ) ) )
7	2, 6	sylbi 121	. . 3 |- ( x e. A -> ( x e. A /\ -. x e. ( B \ A ) ) )
8	1, 7	impbii 126	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
9	8	difeqri 3349	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   \ cdif 3217

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222

This theorem is referenced by:  dif0  3594