Theorem inssdif 3409

Description: Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)

Assertion

Ref	Expression
inssdif	|- ( A i^i B ) C_ ( A \ ( _V \ B ) )

Proof of Theorem inssdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elndif 3297	. . . 4 |- ( x e. B -> -. x e. ( _V \ B ) )
2	1	anim2i 342	. . 3 |- ( ( x e. A /\ x e. B ) -> ( x e. A /\ -. x e. ( _V \ B ) ) )
3		elin 3356	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4		eldif 3175	. . 3 |- ( x e. ( A \ ( _V \ B ) ) <-> ( x e. A /\ -. x e. ( _V \ B ) ) )
5	2, 3, 4	3imtr4i 201	. 2 |- ( x e. ( A i^i B ) -> x e. ( A \ ( _V \ B ) ) )
6	5	ssriv 3197	1 |- ( A i^i B ) C_ ( A \ ( _V \ B ) )

Syntax hints:   -. wn 3   /\ wa 104   e. wcel 2176   _Vcvv 2772   \ cdif 3163   i^i cin 3165   C_ wss 3166

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179

This theorem is referenced by:  difdif2ss  3430