Theorem abn0

Description: Nonempty class abstraction. See also ab0 . (Contributed by NM, 26-Dec-1996) (Proof shortened by Mario Carneiro, 11-Nov-2016) Avoid df-clel , ax-8 . (Revised by Gino Giotto, 30-Aug-2024)

	Ref	Expression
Assertion	abn0	$$⊢\left\{x|\phi \right\}\ne \varnothing \leftrightarrow \exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		ab0	$$⊢\left\{x|\phi \right\}=\varnothing \leftrightarrow \forall x\neg \phi$$
2	1	notbii	$$⊢\neg \left\{x|\phi \right\}=\varnothing \leftrightarrow \neg \forall x\neg \phi$$
3		df-ne	$$⊢\left\{x|\phi \right\}\ne \varnothing \leftrightarrow \neg \left\{x|\phi \right\}=\varnothing$$
4		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
5	2 3 4	3bitr4i	$$⊢\left\{x|\phi \right\}\ne \varnothing \leftrightarrow \exists x\phi$$