Theorem difin2 3421

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 3173	. . . . 5 |- ( A C_ C -> ( x e. A -> x e. C ) )
2	1	pm4.71d 393	. . . 4 |- ( A C_ C -> ( x e. A <-> ( x e. A /\ x e. C ) ) )
3	2	anbi1d 465	. . 3 |- ( A C_ C -> ( ( x e. A /\ -. x e. B ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) ) )
4		eldif 3162	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5		elin 3342	. . . 4 |- ( x e. ( ( C \ B ) i^i A ) <-> ( x e. ( C \ B ) /\ x e. A ) )
6		eldif 3162	. . . . 5 |- ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
7	6	anbi1i 458	. . . 4 |- ( ( x e. ( C \ B ) /\ x e. A ) <-> ( ( x e. C /\ -. x e. B ) /\ x e. A ) )
8		ancom 266	. . . . 5 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
9		anass 401	. . . . 5 |- ( ( ( x e. A /\ x e. C ) /\ -. x e. B ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
10	8, 9	bitr4i 187	. . . 4 |- ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
11	5, 7, 10	3bitri 206	. . 3 |- ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
12	3, 4, 11	3bitr4g 223	. 2 |- ( A C_ C -> ( x e. ( A \ B ) <-> x e. ( ( C \ B ) i^i A ) ) )
13	12	eqrdv 2191	1 |- ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   \ cdif 3150   i^i cin 3152   C_ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166

This theorem is referenced by: (None)