Theorem indif 3447

Description: Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

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

Proof of Theorem indif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anabs5 573	. . 3 |- ( ( x e. A /\ ( x e. A /\ -. x e. B ) ) <-> ( x e. A /\ -. x e. B ) )
2		elin 3387	. . . 4 |- ( x e. ( A i^i ( A \ B ) ) <-> ( x e. A /\ x e. ( A \ B ) ) )
3		eldif 3206	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
4	3	anbi2i 457	. . . 4 |- ( ( x e. A /\ x e. ( A \ B ) ) <-> ( x e. A /\ ( x e. A /\ -. x e. B ) ) )
5	2, 4	bitri 184	. . 3 |- ( x e. ( A i^i ( A \ B ) ) <-> ( x e. A /\ ( x e. A /\ -. x e. B ) ) )
6	1, 5, 3	3bitr4i 212	. 2 |- ( x e. ( A i^i ( A \ B ) ) <-> x e. ( A \ B ) )
7	6	eqriv 2226	1 |- ( A i^i ( A \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1395   e. wcel 2200   \ cdif 3194   i^i cin 3196

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203

This theorem is referenced by:  resdif  5594