Theorem inssdif0im 3536

Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)

Assertion

Ref	Expression
inssdif0im	|- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) )

Proof of Theorem inssdif0im

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3364	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	imbi1i 238	. . . . 5 |- ( ( x e. ( A i^i B ) -> x e. C ) <-> ( ( x e. A /\ x e. B ) -> x e. C ) )
3		imanim 690	. . . . 5 |- ( ( ( x e. A /\ x e. B ) -> x e. C ) -> -. ( ( x e. A /\ x e. B ) /\ -. x e. C ) )
4	2, 3	sylbi 121	. . . 4 |- ( ( x e. ( A i^i B ) -> x e. C ) -> -. ( ( x e. A /\ x e. B ) /\ -. x e. C ) )
5		eldif 3183	. . . . . 6 |- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
6	5	anbi2i 457	. . . . 5 |- ( ( x e. A /\ x e. ( B \ C ) ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
7		elin 3364	. . . . 5 |- ( x e. ( A i^i ( B \ C ) ) <-> ( x e. A /\ x e. ( B \ C ) ) )
8		anass 401	. . . . 5 |- ( ( ( x e. A /\ x e. B ) /\ -. x e. C ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
9	6, 7, 8	3bitr4ri 213	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ -. x e. C ) <-> x e. ( A i^i ( B \ C ) ) )
10	4, 9	sylnib 678	. . 3 |- ( ( x e. ( A i^i B ) -> x e. C ) -> -. x e. ( A i^i ( B \ C ) ) )
11	10	alimi 1479	. 2 |- ( A. x ( x e. ( A i^i B ) -> x e. C ) -> A. x -. x e. ( A i^i ( B \ C ) ) )
12		ssalel 3189	. 2 |- ( ( A i^i B ) C_ C <-> A. x ( x e. ( A i^i B ) -> x e. C ) )
13		eq0 3487	. 2 |- ( ( A i^i ( B \ C ) ) = (/) <-> A. x -. x e. ( A i^i ( B \ C ) ) )
14	11, 12, 13	3imtr4i 201	1 |- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   e. wcel 2178   \ cdif 3171   i^i cin 3173   C_ wss 3174   (/)c0 3468

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by:  disjdif  3541