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Metamath Proof Explorer

Theorem difin

Description: Difference with intersection. Theorem 33 of Suppes p. 29. (Contributed by NM, 31-Mar-1998) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Assertion difin 
|- ( A \ ( A i^i B ) ) = ( A \ B )

Proof

Step Hyp Ref Expression
1 pm4.61 
 |-  ( -. ( x e. A -> x e. B ) <-> ( x e. A /\ -. x e. B ) )
2 anclb 
 |-  ( ( x e. A -> x e. B ) <-> ( x e. A -> ( x e. A /\ x e. B ) ) )
3 elin 
 |-  ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4 3 imbi2i 
 |-  ( ( x e. A -> x e. ( A i^i B ) ) <-> ( x e. A -> ( x e. A /\ x e. B ) ) )
5 iman 
 |-  ( ( x e. A -> x e. ( A i^i B ) ) <-> -. ( x e. A /\ -. x e. ( A i^i B ) ) )
6 2 4 5 3bitr2i 
 |-  ( ( x e. A -> x e. B ) <-> -. ( x e. A /\ -. x e. ( A i^i B ) ) )
7 6 con2bii 
 |-  ( ( x e. A /\ -. x e. ( A i^i B ) ) <-> -. ( x e. A -> x e. B ) )
8 eldif 
 |-  ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
9 1 7 8 3bitr4i 
 |-  ( ( x e. A /\ -. x e. ( A i^i B ) ) <-> x e. ( A \ B ) )
10 9 difeqri 
 |-  ( A \ ( A i^i B ) ) = ( A \ B )