Theorem inssdif 3317

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 3205	. . . 4 |- ( x e. B -> -. x e. ( _V \ B ) )
2	1	anim2i 340	. . 3 |- ( ( x e. A /\ x e. B ) -> ( x e. A /\ -. x e. ( _V \ B ) ) )
3		elin 3264	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4		eldif 3085	. . 3 |- ( x e. ( A \ ( _V \ B ) ) <-> ( x e. A /\ -. x e. ( _V \ B ) ) )
5	2, 3, 4	3imtr4i 200	. 2 |- ( x e. ( A i^i B ) -> x e. ( A \ ( _V \ B ) ) )
6	5	ssriv 3106	1 |- ( A i^i B ) C_ ( A \ ( _V \ B ) )

Syntax hints:   -. wn 3   /\ wa 103   e. wcel 1481   _Vcvv 2689   \ cdif 3073   i^i cin 3075   C_ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089

This theorem is referenced by:  difdif2ss  3338