Theorem inssdif 3358

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 3246	. . . 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 3305	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4		eldif 3125	. . 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 3146	1 |- ( A i^i B ) C_ ( A \ ( _V \ B ) )

Syntax hints:   -. wn 3   /\ wa 103   e. wcel 2136   _Vcvv 2726   \ cdif 3113   i^i cin 3115   C_ wss 3116

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129

This theorem is referenced by:  difdif2ss  3379