Theorem difin2 3389

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 3141	. . . . 5 |- ( A C_ C -> ( x e. A -> x e. C ) )
2	1	pm4.71d 391	. . . 4 |- ( A C_ C -> ( x e. A <-> ( x e. A /\ x e. C ) ) )
3	2	anbi1d 462	. . 3 |- ( A C_ C -> ( ( x e. A /\ -. x e. B ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) ) )
4		eldif 3130	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3310	. . . 4 |- ( x e. ( ( C \ B ) i^i A ) <-> ( x e. ( C \ B ) /\ x e. A ) )
6		eldif 3130	. . . . 5 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
7	6	anbi1i 455	. . . 4 |- ( ( x e. ( C \ B ) /\ x e. A ) <-> ( ( x e. C /\ -. x e. B ) /\ x e. A ) )
8		ancom 264	. . . . 5 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
9		anass 399	. . . . 5 |- ( ( ( x e. A /\ x e. C ) /\ -. x e. B ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
10	8, 9	bitr4i 186	. . . 4 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
11	5, 7, 10	3bitri 205	. . 3 |- ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
12	3, 4, 11	3bitr4g 222	. 2 |- ( A C_ C -> ( x e. ( A \ B ) <-> x e. ( ( C \ B ) i^i A ) ) )
13	12	eqrdv 2168	1 |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   \ cdif 3118   i^i cin 3120   C_ wss 3121

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134

This theorem is referenced by: (None)