Theorem indif 3474

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 579	. . 3 |- ( ( x e. A /\ ( x e. A /\ -. x e. B ) ) <-> ( x e. A /\ -. x e. B ) )
2		elin 3412	. . . 4 |- ( x e. ( A i^i ( A \ B ) ) <-> ( x e. A /\ x e. ( A \ B ) ) )
3		eldif 3229	. . . . 5 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
4	3	anbi2i 461	. . . 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 2235	1 |- ( A i^i ( A \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1402   e. wcel 2209   \ cdif 3217   i^i cin 3219

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  df-in 3226

This theorem is referenced by:  resdif  5656