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Metamath Proof Explorer

Theorem difin2

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

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

Proof

Step Hyp Ref Expression
1 ssel 
 |-  ( A C_ C -> ( x e. A -> x e. C ) )
2 1 pm4.71d 
 |-  ( A C_ C -> ( x e. A <-> ( x e. A /\ x e. C ) ) )
3 2 anbi1d 
 |-  ( A C_ C -> ( ( x e. A /\ -. x e. B ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) ) )
4 eldif 
 |-  ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5 ancom 
 |-  ( ( ( x e. C /\ -. x e. B ) /\ x e. A ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
6 elin 
 |-  ( x e. ( ( C \ B ) i^i A ) <-> ( x e. ( C \ B ) /\ x e. A ) )
7 eldif 
 |-  ( x e. ( C \ B ) <-> ( x e. C /\ -. x e. B ) )
8 6 7 bianbi 
 |-  ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. C /\ -. x e. B ) /\ x e. A ) )
9 anass 
 |-  ( ( ( x e. A /\ x e. C ) /\ -. x e. B ) <-> ( x e. A /\ ( x e. C /\ -. x e. B ) ) )
10 5 8 9 3bitr4i 
 |-  ( x e. ( ( C \ B ) i^i A ) <-> ( ( x e. A /\ x e. C ) /\ -. x e. B ) )
11 3 4 10 3bitr4g 
 |-  ( A C_ C -> ( x e. ( A \ B ) <-> x e. ( ( C \ B ) i^i A ) ) )
12 11 eqrdv 
 |-  ( A C_ C -> ( A \ B ) = ( ( C \ B ) i^i A ) )