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Metamath Proof Explorer

Theorem mpo0v

Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015) (Revised by Mario Carneiro, 15-May-2015) (Proof shortened by AV, 27-Jan-2024)

		Ref	Expression
	Assertion	mpo0v	$⊢ (x \in \emptyset, y \in B ⟼ C) = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		eqid	$⊢ \emptyset = \emptyset$
2	1	orci	$⊢ (\emptyset = \emptyset \lor B = \emptyset)$
3		0mpo0	$⊢ (\emptyset = \emptyset \lor B = \emptyset) \to (x \in \emptyset, y \in B ⟼ C) = \emptyset$
4	2 3	ax-mp	$⊢ (x \in \emptyset, y \in B ⟼ C) = \emptyset$