Theorem indif 3360

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 563	. . 3 |- ( ( x e. A /\ ( x e. A /\ -. x e. B ) ) <-> ( x e. A /\ -. x e. B ) )
2		elin 3300	. . . 4 |- ( x e. ( A i^i ( A \ B ) ) <-> ( x e. A /\ x e. ( A \ B ) ) )
3		eldif 3120	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
4	3	anbi2i 453	. . . 4 |- ( ( x e. A /\ x e. ( A \ B ) ) <-> ( x e. A /\ ( x e. A /\ -. x e. B ) ) )
5	2, 4	bitri 183	. . 3 |- ( x e. ( A i^i ( A \ B ) ) <-> ( x e. A /\ ( x e. A /\ -. x e. B ) ) )
6	1, 5, 3	3bitr4i 211	. 2 |- ( x e. ( A i^i ( A \ B ) ) <-> x e. ( A \ B ) )
7	6	eqriv 2161	1 |- ( A i^i ( A \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1342   e. wcel 2135   \ cdif 3108   i^i cin 3110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-dif 3113  df-in 3117

This theorem is referenced by:  resdif  5448