Theorem difin2 3226

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 2993	. . . . 5 |- ( A C_ C -> ( x e. A -> x e. C ) )
2	1	pm4.71d 385	. . . 4 |- ( A C_ C -> ( x e. A <-> ( x e. A /\ x e. C ) ) )
3	2	anbi1d 452	. . 3 |- ( A C_ C -> ( ( x e. A /\ -. x e. B ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) ) )
4		eldif 2982	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3155	. . . 4 |- ( x e. ( ( C \ B ) i^i A ) <-> ( x e. ( C \ B ) /\ x e. A ) )
6		eldif 2982	. . . . 5 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
7	6	anbi1i 445	. . . 4 |- ( ( x e. ( C \ B ) /\ x e. A ) <-> ( ( x e. C /\ -. x e. B ) /\ x e. A ) )
8		ancom 262	. . . . 5 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
9		anass 393	. . . . 5 |- ( ( ( x e. A /\ x e. C ) /\ -. x e. B ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
10	8, 9	bitr4i 185	. . . 4 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
11	5, 7, 10	3bitri 204	. . 3 |- ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
12	3, 4, 11	3bitr4g 221	. 2 |- ( A C_ C -> ( x e. ( A \ B ) <-> x e. ( ( C \ B ) i^i A ) ) )
13	12	eqrdv 2079	1 |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   \ cdif 2970   i^i cin 2972   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986

This theorem is referenced by: (None)