Theorem bj-0nelmpt

Description: The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020)

	Ref	Expression
Assertion	bj-0nelmpt	$$⊢\neg \varnothing \in \left(x\in A⟼B\right)$$

Proof

Step	Hyp	Ref	Expression
1		0nelopab	$$⊢\neg \varnothing \in \left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}$$
2		df-mpt	$$⊢\left(x\in A⟼B\right)=\left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}$$
3	2	eqcomi	$$⊢\left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}=\left(x\in A⟼B\right)$$
4	3	eleq2i	$$⊢\varnothing \in \left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}\leftrightarrow \varnothing \in \left(x\in A⟼B\right)$$
5	1 4	mtbi	$$⊢\neg \varnothing \in \left(x\in A⟼B\right)$$