Theorem inssdif 3363

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 3251	. . . 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 3310	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4		eldif 3130	. . 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 3151	1 |- ( A i^i B ) C_ ( A \ ( _V \ B ) )

Syntax hints:   -. wn 3   /\ wa 103   e. wcel 2141   _Vcvv 2730   \ cdif 3118   i^i cin 3120   C_ wss 3121

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134

This theorem is referenced by:  difdif2ss  3384