Theorem bj-0nelopab

Description: The empty set is not an element of 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	$$⊢\neg \varnothing \in \left\{⟨x,y⟩|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		relopab	$$⊢\mathrm{Rel}\left\{⟨x,y⟩|\phi \right\}$$
2		0nelrel0	$$⊢\mathrm{Rel}\left\{⟨x,y⟩|\phi \right\}\to \neg \varnothing \in \left\{⟨x,y⟩|\phi \right\}$$
3	1 2	ax-mp	$$⊢\neg \varnothing \in \left\{⟨x,y⟩|\phi \right\}$$