Theorem invdif 3179

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 2560	. . . . 5 |- x e. _V
2		eldif 2927	. . . . 5 |- ( x e. ( _V \ B ) <-> ( x e. _V /\ -. x e. B ) )
3	1, 2	mpbiran 847	. . . 4 |- ( x e. ( _V \ B ) <-> -. x e. B )
4	3	anbi2i 430	. . 3 |- ( ( x e. A /\ x e. ( _V \ B ) ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3126	. . 3 |- ( x e. ( A i^i ( _V \ B ) ) <-> ( x e. A /\ x e. ( _V \ B ) ) )
6		eldif 2927	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
7	4, 5, 6	3bitr4i 201	. 2 |- ( x e. ( A i^i ( _V \ B ) ) <-> x e. ( A \ B ) )
8	7	eqriv 2037	1 |- ( A i^i ( _V \ B ) ) = ( A \ B )

Syntax hints:   -. wn 3   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   \ cdif 2914   i^i cin 2916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924

This theorem is referenced by:  indif2  3181  difundir  3190  difindir  3192  difdif2ss  3194  difun1  3197  difdifdirss  3307  nn0supp  8234