Theorem inssdif0 3618

Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
inssdif0	|- ((A i^i B) C_ C <-> (A i^i (B \ C)) = (/))

Proof of Theorem inssdif0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3220	. . . . . 6 |- (x e. (A i^i B) <-> (x e. A /\ x e. B))
2	1	imbi1i 315	. . . . 5 |- ((x e. (A i^i B) -> x e. C) <-> ((x e. A /\ x e. B) -> x e. C))
3		iman 413	. . . . 5 |- (((x e. A /\ x e. B) -> x e. C) <-> -. ((x e. A /\ x e. B) /\ -. x e. C))
4	2, 3	bitri 240	. . . 4 |- ((x e. (A i^i B) -> x e. C) <-> -. ((x e. A /\ x e. B) /\ -. x e. C))
5		eldif 3222	. . . . . 6 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
6	5	anbi2i 675	. . . . 5 |- ((x e. A /\ x e. (B \ C)) <-> (x e. A /\ (x e. B /\ -. x e. C)))
7		elin 3220	. . . . 5 |- (x e. (A i^i (B \ C)) <-> (x e. A /\ x e. (B \ C)))
8		anass 630	. . . . 5 |- (((x e. A /\ x e. B) /\ -. x e. C) <-> (x e. A /\ (x e. B /\ -. x e. C)))
9	6, 7, 8	3bitr4ri 269	. . . 4 |- (((x e. A /\ x e. B) /\ -. x e. C) <-> x e. (A i^i (B \ C)))
10	4, 9	xchbinx 301	. . 3 |- ((x e. (A i^i B) -> x e. C) <-> -. x e. (A i^i (B \ C)))
11	10	albii 1566	. 2 |- (A.x(x e. (A i^i B) -> x e. C) <-> A.x -. x e. (A i^i (B \ C)))
12		dfss2 3263	. 2 |- ((A i^i B) C_ C <-> A.x(x e. (A i^i B) -> x e. C))
13		eq0 3565	. 2 |- ((A i^i (B \ C)) = (/) <-> A.x -. x e. (A i^i (B \ C)))
14	11, 12, 13	3bitr4i 268	1 |- ((A i^i B) C_ C <-> (A i^i (B \ C)) = (/))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   \ cdif 3207   i^i cin 3209   C_ wss 3258  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  disjdif  3623