Theorem inssdif0im 3425

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 3254	. . . . . 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 677	. . . . 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 3075	. . . . . 6 |- ( x e. ( B \ C ) <-> ( x e. B /\ -. x e. C ) )
6	5	anbi2i 452	. . . . 5 |- ( ( x e. A /\ x e. ( B \ C ) ) <-> ( x e. A /\ ( x e. B /\ -. x e. C ) ) )
7		elin 3254	. . . . 5 |- ( x e. ( A i^i ( B \ C ) ) <-> ( x e. A /\ x e. ( B \ C ) ) )
8		anass 398	. . . . 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 665	. . 3 |- ( ( x e. ( A i^i B ) -> x e. C ) -> -. x e. ( A i^i ( B \ C ) ) )
11	10	alimi 1431	. 2 |- ( A. x ( x e. ( A i^i B ) -> x e. C ) -> A. x -. x e. ( A i^i ( B \ C ) ) )
12		dfss2 3081	. 2 |- ( ( A i^i B ) C_ C <-> A. x ( x e. ( A i^i B ) -> x e. C ) )
13		eq0 3376	. 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 1329   = wceq 1331   e. wcel 1480   \ cdif 3063   i^i cin 3065   C_ wss 3066   (/)c0 3358

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359

This theorem is referenced by:  disjdif  3430