Theorem ab0

Description: The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne ). (Contributed by BJ, 19-Mar-2021)

	Ref	Expression
Assertion	ab0	$$⊢\left\{x|\phi \right\}=\varnothing \leftrightarrow \forall x\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		nfab1	$$⊢\underset{\_}{Ⅎ}x\left\{x|\phi \right\}$$
2	1	eq0f	$$⊢\left\{x|\phi \right\}=\varnothing \leftrightarrow \forall x\neg x\in \left\{x|\phi \right\}$$
3		abid	$$⊢x\in \left\{x|\phi \right\}\leftrightarrow \phi$$
4	3	notbii	$$⊢\neg x\in \left\{x|\phi \right\}\leftrightarrow \neg \phi$$
5	4	albii	$$⊢\forall x\neg x\in \left\{x|\phi \right\}\leftrightarrow \forall x\neg \phi$$
6	2 5	bitri	$$⊢\left\{x|\phi \right\}=\varnothing \leftrightarrow \forall x\neg \phi$$