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 Gino Giotto and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 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		dfcleq	$$⊢A=\left\{y|\perp \right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in \left\{y|\perp \right\}\right)$$
4		df-clab	$$⊢x\in \left\{y|\perp \right\}\leftrightarrow \left[x/y\right]\perp$$
5		sbv	$$⊢\left[x/y\right]\perp \leftrightarrow \perp$$
6	4 5	bitri	$$⊢x\in \left\{y|\perp \right\}\leftrightarrow \perp$$
7	6	bibi2i	$$⊢\left(x\in A\leftrightarrow x\in \left\{y|\perp \right\}\right)\leftrightarrow \left(x\in A\leftrightarrow \perp \right)$$
8		nbfal	$$⊢\neg x\in A\leftrightarrow \left(x\in A\leftrightarrow \perp \right)$$
9	7 8	bitr4i	$$⊢\left(x\in A\leftrightarrow x\in \left\{y|\perp \right\}\right)\leftrightarrow \neg x\in A$$
10	9	albii	$$⊢\forall x\left(x\in A\leftrightarrow x\in \left\{y|\perp \right\}\right)\leftrightarrow \forall x\neg x\in A$$
11	3 10	bitri	$$⊢A=\left\{y|\perp \right\}\leftrightarrow \forall x\neg x\in A$$
12	2 11	bitri	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$