Theorem inssdif0im 3285

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 3120	. . . . . 6 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
2	1	imbi1i 227	. . . . 5 |- ((x e. (A i^i B) -> x e. C) <-> ((x e. A /\ x e. B) -> x e. C))
3		imanim 784	. . . . 5 |- (((x e. A /\ x e. B) -> x e. C) -> -. ((x e. A /\ x e. B) /\ -. x e. C))
4	2, 3	sylbi 114	. . . 4 |- ((x e. (A i^i B) -> x e. C) -> -. ((x e. A /\ x e. B) /\ -. x e. C))
5		eldif 2921	. . . . . 6 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
6	5	anbi2i 430	. . . . 5 |- ((x e. A /\ x e. (B \ C)) <-> (x e. A /\ (x e. B /\ -. x e. C)))
7		elin 3120	. . . . 5 |- (x e. (A i^i (B \ C)) <-> (x e. A /\ x e. (B \ C)))
8		anass 381	. . . . 5 |- (((x e. A /\ x e. B) /\ -. x e. C) <-> (x e. A /\ (x e. B /\ -. x e. C)))
9	6, 7, 8	3bitr4ri 202	. . . 4 |- (((x e. A /\ x e. B) /\ -. x e. C) <-> x e. (A i^i (B \ C)))
10	4, 9	sylnib 600	. . 3 |- ((x e. (A i^i B) -> x e. C) -> -. x e. (A i^i (B \ C)))
11	10	alimi 1341	. 2 |- (A.x(x e. (A i^i B) -> x e. C) -> A.x -. x e. (A i^i (B \ C)))
12		dfss2 2928	. 2 |- ((A i^i B) C_ C <-> A.x(x e. (A i^i B) -> x e. C))
13		eq0 3233	. 2 |- ((A i^i (B \ C)) = (/) <-> A.x -. x e. (A i^i (B \ C)))
14	11, 12, 13	3imtr4i 190	1 |- ((A i^i B) C_ C -> (A i^i (B \ C)) = (/))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97  A.wal 1240   = wceq 1242   e. wcel 1390   \ cdif 2908   i^i cin 2910   C_ wss 2911   (/)c0 3218

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219

This theorem is referenced by:  disjdif  3290