Theorem inssdif0im 3457

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 3286	. . . . . 6 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	imbi1i 237	. . . . 5 |- ( ( x e. ( A i^i B ) -> x e. C ) <-> ( ( x e. A /\ x e. B ) -> x e. C ) )
3		imanim 678	. . . . 5 |- ( ( ( x e. A /\ x e. B ) -> x e. C ) -> -. ( ( x e. A /\ x e. B ) /\ -. x e. C ) )
4	2, 3	sylbi 120	. . . 4 |- ( ( x e. ( A i^i B ) -> x e. C ) -> -. ( ( x e. A /\ x e. B ) /\ -. x e. C ) )
5		eldif 3107	. . . . . 6 |- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
6	5	anbi2i 453	. . . . 5 |- ( ( x e. A /\ x e. ( B \ C ) ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
7		elin 3286	. . . . 5 |- ( x e. ( A i^i ( B \ C ) ) <-> ( x e. A /\ x e. ( B \ C ) ) )
8		anass 399	. . . . 5 |- ( ( ( x e. A /\ x e. B ) /\ -. x e. C ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
9	6, 7, 8	3bitr4ri 212	. . . 4 |- ( ( ( x e. A /\ x e. B ) /\ -. x e. C ) <-> x e. ( A i^i ( B \ C ) ) )
10	4, 9	sylnib 666	. . 3 |- ( ( x e. ( A i^i B ) -> x e. C ) -> -. x e. ( A i^i ( B \ C ) ) )
11	10	alimi 1432	. 2 |- ( A. x ( x e. ( A i^i B ) -> x e. C ) -> A. x -. x e. ( A i^i ( B \ C ) ) )
12		dfss2 3113	. 2 |- ( ( A i^i B ) C_ C <-> A. x ( x e. ( A i^i B ) -> x e. C ) )
13		eq0 3408	. 2 |- ( ( A i^i ( B \ C ) ) = (/) <-> A. x -. x e. ( A i^i ( B \ C ) ) )
14	11, 12, 13	3imtr4i 200	1 |- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1330   = wceq 1332   e. wcel 2125   \ cdif 3095   i^i cin 3097   C_ wss 3098   (/)c0 3390

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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-nul 3391

This theorem is referenced by:  disjdif  3462