Theorem difin2 3517

Description: Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
difin2	|- (A C_ C -> (A \ B) = ((C \ B) i^i A))

Proof of Theorem difin2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . 5 |- (A C_ C -> (x e. A -> x e. C))
2	1	pm4.71d 615	. . . 4 |- (A C_ C -> (x e. A <-> (x e. A /\ x e. C)))
3	2	anbi1d 685	. . 3 |- (A C_ C -> ((x e. A /\ -. x e. B) <-> ((x e. A /\ x e. C) /\ -. x e. B)))
4		eldif 3222	. . 3 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
5		elin 3220	. . . 4 |- (x e. ((C \ B) i^i A) <-> (x e. (C \ B) /\ x e. A))
6		eldif 3222	. . . . 5 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
7	6	anbi1i 676	. . . 4 |- ((x e. (C \ B) /\ x e. A) <-> ((x e. C /\ -. x e. B) /\ x e. A))
8		ancom 437	. . . . 5 |- (((x e. C /\ -. x e. B) /\ x e. A) <-> (x e. A /\ (x e. C /\ -. x e. B)))
9		anass 630	. . . . 5 |- (((x e. A /\ x e. C) /\ -. x e. B) <-> (x e. A /\ (x e. C /\ -. x e. B)))
10	8, 9	bitr4i 243	. . . 4 |- (((x e. C /\ -. x e. B) /\ x e. A) <-> ((x e. A /\ x e. C) /\ -. x e. B))
11	5, 7, 10	3bitri 262	. . 3 |- (x e. ((C \ B) i^i A) <-> ((x e. A /\ x e. C) /\ -. x e. B))
12	3, 4, 11	3bitr4g 279	. 2 |- (A C_ C -> (x e. (A \ B) <-> x e. ((C \ B) i^i A)))
13	12	eqrdv 2351	1 |- (A C_ C -> (A \ B) = ((C \ B) i^i A))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   \ cdif 3207   i^i cin 3209   C_ wss 3258

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by: (None)