Theorem disjdifprg2

Description: A trivial partition of a set into its difference and intersection with another set. (Contributed by Thierry Arnoux, 25-Dec-2016)

	Ref	Expression
Assertion	disjdifprg2	|- ( A e. V -> Disj_ x e. { ( A \ B ) , ( A i^i B ) } x )

Proof

Step	Hyp	Ref	Expression
1		inex1g	|- ( A e. V -> ( A i^i B ) e. _V )
2		elex	|- ( A e. V -> A e. _V )
3		disjdifprg	|- ( ( ( A i^i B ) e. _V /\ A e. _V ) -> Disj_ x e. { ( A \ ( A i^i B ) ) , ( A i^i B ) } x )
4	1 2 3	syl2anc	|- ( A e. V -> Disj_ x e. { ( A \ ( A i^i B ) ) , ( A i^i B ) } x )
5		difin	|- ( A \ ( A i^i B ) ) = ( A \ B )
6	5	preq1i	|- { ( A \ ( A i^i B ) ) , ( A i^i B ) } = { ( A \ B ) , ( A i^i B ) }
7	6	a1i	|- ( A e. V -> { ( A \ ( A i^i B ) ) , ( A i^i B ) } = { ( A \ B ) , ( A i^i B ) } )
8	7	disjeq1d	|- ( A e. V -> ( Disj_ x e. { ( A \ ( A i^i B ) ) , ( A i^i B ) } x <-> Disj_ x e. { ( A \ B ) , ( A i^i B ) } x ) )
9	4 8	mpbid	|- ( A e. V -> Disj_ x e. { ( A \ B ) , ( A i^i B ) } x )