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Metamath Proof Explorer

Theorem xp0

Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of Monk1 p. 37. (Contributed by NM, 12-Apr-2004) Avoid axioms. (Revised by TM, 1-Feb-2026)

		Ref	Expression
	Assertion	xp0	$⊢ A \times \emptyset = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		noel	$⊢ \neg y \in \emptyset$
2		simprr	$⊢ (z = ⟨x, y⟩ \land (x \in A \land y \in \emptyset)) \to y \in \emptyset$
3	1 2	mto	$⊢ \neg (z = ⟨x, y⟩ \land (x \in A \land y \in \emptyset))$
4	3	nex	$⊢ \neg \exists y (z = ⟨x, y⟩ \land (x \in A \land y \in \emptyset))$
5	4	nex	$⊢ \neg \exists x \exists y (z = ⟨x, y⟩ \land (x \in A \land y \in \emptyset))$
6		elxpi	$⊢ z \in (A \times \emptyset) \to \exists x \exists y (z = ⟨x, y⟩ \land (x \in A \land y \in \emptyset))$
7	5 6	mto	$⊢ \neg z \in (A \times \emptyset)$
8	7	nel0	$⊢ A \times \emptyset = \emptyset$