Theorem xpcoidgend

Description: If two classes are not disjoint, then the composition of their Cartesian product with itself is idempotent. (Contributed by RP, 24-Dec-2019)

	Ref	Expression
Hypothesis	xpcoidgend.1	|- ( ph -> ( A i^i B ) =/= (/) )
Assertion	xpcoidgend	|- ( ph -> ( ( A X. B ) o. ( A X. B ) ) = ( A X. B ) )

Proof

Step	Hyp	Ref	Expression
1		xpcoidgend.1	|- ( ph -> ( A i^i B ) =/= (/) )
2		incom	|- ( A i^i B ) = ( B i^i A )
3	2 1	eqnetrrid	|- ( ph -> ( B i^i A ) =/= (/) )
4	3	xpcogend	|- ( ph -> ( ( A X. B ) o. ( A X. B ) ) = ( A X. B ) )