Theorem bj-0nelopab

Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017) (Proof shortened by BJ, 22-Jul-2023)

TODO: move to the main section when one can reorder sections so that we can use relopab (this is a very limited reordering).

	Ref	Expression
Assertion	bj-0nelopab	⊢ ¬ ∅ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		relopab	⊢ Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
2		0nelrel0	⊢ ( Rel { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } → ¬ ∅ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
3	1 2	ax-mp	⊢ ¬ ∅ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }