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Metamath Proof Explorer

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	$⊢ φ \to A \cap B \neq \emptyset$
	Assertion	xpcoidgend	$⊢ φ \to (A \times B) \circ (A \times B) = A \times B$

Proof

Step	Hyp	Ref	Expression
1		xpcoidgend.1	$⊢ φ \to A \cap B \neq \emptyset$
2		incom	$⊢ A \cap B = B \cap A$
3	2 1	eqnetrrid	$⊢ φ \to B \cap A \neq \emptyset$
4	3	xpcogend	$⊢ φ \to (A \times B) \circ (A \times B) = A \times B$