Theorem eq0

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by GG, 6-Sep-2024)

	Ref	Expression
Assertion	eq0	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$

Proof

Step	Hyp	Ref	Expression
1		dfnul4	$$⊢\varnothing =\left\{y|\perp \right\}$$
2	1	eqeq2i	$$⊢A=\varnothing \leftrightarrow A=\left\{y|\perp \right\}$$
3		biidd	$$⊢y=x\to \left(\perp \leftrightarrow \perp \right)$$
4	3	eqabbw	$$⊢A=\left\{y|\perp \right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow \perp \right)$$
5		nbfal	$$⊢\neg x\in A\leftrightarrow \left(x\in A\leftrightarrow \perp \right)$$
6	5	albii	$$⊢\forall x\neg x\in A\leftrightarrow \forall x\left(x\in A\leftrightarrow \perp \right)$$
7	4 6	bitr4i	$$⊢A=\left\{y|\perp \right\}\leftrightarrow \forall x\neg x\in A$$
8	2 7	bitri	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$