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	⊢ ¬ ∅ ∈ ( 𝑥 ∈ 𝐴 ↦ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		0nelopab	⊢ ¬ ∅ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
2		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
3	2	eqcomi	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) } = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
4	3	eleq2i	⊢ ( ∅ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) } ↔ ∅ ∈ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) )
5	1 4	mtbi	⊢ ¬ ∅ ∈ ( 𝑥 ∈ 𝐴 ↦ 𝐵 )