Theorem inssdif 3395

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 3283	. . . 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 3342	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4		eldif 3162	. . 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 3183	1 |- ( A i^i B ) C_ ( A \ ( _V \ B ) )

Syntax hints:   -. wn 3   /\ wa 104   e. wcel 2164   _Vcvv 2760   \ cdif 3150   i^i cin 3152   C_ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166

This theorem is referenced by:  difdif2ss  3416