Theorem invdif 3473

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 2824	. . . . 5 |- x e. _V
2		eldif 3229	. . . . 5 |- ( x e. ( _V \ B ) <-> ( x e. _V /\ -. x e. B ) )
3	1, 2	mpbiran 953	. . . 4 |- ( x e. ( _V \ B ) <-> -. x e. B )
4	3	anbi2i 461	. . 3 |- ( ( x e. A /\ x e. ( _V \ B ) ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3412	. . 3 |- ( x e. ( A i^i ( _V \ B ) ) <-> ( x e. A /\ x e. ( _V \ B ) ) )
6		eldif 3229	. . 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 2235	1 |- ( A i^i ( _V \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   \ 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:  indif2  3475  difundir  3484  difindir  3486  difdif2ss  3488  difun1  3491  difdifdirss  3609  fsuppeq  6477  fsuppeqg  6478  nn0supp  9598