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	$$⊢\left(x\in \varnothing ,y\in B⟼C\right)=\varnothing$$

Proof

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