Theorem inssdif0

Description: Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996) (Proof shortened by Andrew Salmon, 26-Jun-2011) (Proof shortened by Wolf Lammen, 30-Sep-2014) (Proof shortened by BJ, 18-Jul-2026)

	Ref	Expression
Assertion	inssdif0	|- ( ( A i^i B ) C_ C <-> ( A i^i ( B \ C ) ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		ssdif0	|- ( ( A i^i B ) C_ C <-> ( ( A i^i B ) \ C ) = (/) )
2		indif2	|- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )
3	2	eqeq1i	|- ( ( A i^i ( B \ C ) ) = (/) <-> ( ( A i^i B ) \ C ) = (/) )
4	1 3	bitr4i	|- ( ( A i^i B ) C_ C <-> ( A i^i ( B \ C ) ) = (/) )