Theorem invdif 3224

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 2615	. . . . 5 |- x e. _V
2		eldif 2993	. . . . 5 |- ( x e. ( _V \ B ) <-> ( x e. _V /\ -. x e. B ) )
3	1, 2	mpbiran 882	. . . 4 |- ( x e. ( _V \ B ) <-> -. x e. B )
4	3	anbi2i 445	. . 3 |- ( ( x e. A /\ x e. ( _V \ B ) ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3167	. . 3 |- ( x e. ( A i^i ( _V \ B ) ) <-> ( x e. A /\ x e. ( _V \ B ) ) )
6		eldif 2993	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
7	4, 5, 6	3bitr4i 210	. 2 |- ( x e. ( A i^i ( _V \ B ) ) <-> x e. ( A \ B ) )
8	7	eqriv 2080	1 |- ( A i^i ( _V \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1285   e. wcel 1434   _Vcvv 2612   \ cdif 2981   i^i cin 2983

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-in 2990

This theorem is referenced by:  indif2  3226  difundir  3235  difindir  3237  difdif2ss  3239  difun1  3242  difdifdirss  3348  nn0supp  8615