Theorem invdif 3423

Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)

Assertion

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

Proof of Theorem invdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2779	. . . . 5 |- x e. _V
2		eldif 3183	. . . . 5 |- ( x e. ( _V \ B ) <-> ( x e. _V /\ -. x e. B ) )
3	1, 2	mpbiran 943	. . . 4 |- ( x e. ( _V \ B ) <-> -. x e. B )
4	3	anbi2i 457	. . 3 |- ( ( x e. A /\ x e. ( _V \ B ) ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3364	. . 3 |- ( x e. ( A i^i ( _V \ B ) ) <-> ( x e. A /\ x e. ( _V \ B ) ) )
6		eldif 3183	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
7	4, 5, 6	3bitr4i 212	. 2 |- ( x e. ( A i^i ( _V \ B ) ) <-> x e. ( A \ B ) )
8	7	eqriv 2204	1 |- ( A i^i ( _V \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   \ cdif 3171   i^i cin 3173

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180

This theorem is referenced by:  indif2  3425  difundir  3434  difindir  3436  difdif2ss  3438  difun1  3441  difdifdirss  3553  nn0supp  9382