Theorem invdif 3414

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 2774	. . . . 5 |- x e. _V
2		eldif 3174	. . . . 5 |- ( x e. ( _V \ B ) <-> ( x e. _V /\ -. x e. B ) )
3	1, 2	mpbiran 942	. . . 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 3355	. . 3 |- ( x e. ( A i^i ( _V \ B ) ) <-> ( x e. A /\ x e. ( _V \ B ) ) )
6		eldif 3174	. . 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 2201	1 |- ( A i^i ( _V \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   \ cdif 3162   i^i cin 3164

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171

This theorem is referenced by:  indif2  3416  difundir  3425  difindir  3427  difdif2ss  3429  difun1  3432  difdifdirss  3544  nn0supp  9346