0xp

Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of Monk1 p. 37. (Contributed by NM, 4-Jul-1994)

		Ref	Expression
	Assertion	0xp	$⊢ \emptyset \times A = \emptyset$

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

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