Theorem undisj2

Description: The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004)

	Ref	Expression
Assertion	undisj2	|- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		un00	|- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
2		indi	|- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
3	2	eqeq1i	|- ( ( A i^i ( B u. C ) ) = (/) <-> ( ( A i^i B ) u. ( A i^i C ) ) = (/) )
4	1 3	bitr4i	|- ( ( ( A i^i B ) = (/) /\ ( A i^i C ) = (/) ) <-> ( A i^i ( B u. C ) ) = (/) )