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)

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

Proof

Step	Hyp	Ref	Expression
1		dfcleq	$$⊢A=\varnothing \leftrightarrow \forall x\left(x\in A\leftrightarrow x\in \varnothing \right)$$
2		noel	$$⊢\neg x\in \varnothing$$
3	2	nbn	$$⊢\neg x\in A\leftrightarrow \left(x\in A\leftrightarrow x\in \varnothing \right)$$
4	3	albii	$$⊢\forall x\neg x\in A\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in \varnothing \right)$$
5	1 4	bitr4i	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$