Theorem neq0

Description: A class is not empty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. (Contributed by NM, 21-Jun-1993) Avoid ax-11 , ax-12 . (Revised by Gino Giotto, 28-Jun-2024)

	Ref	Expression
Assertion	neq0	$$⊢\neg A=\varnothing \leftrightarrow \exists xx\in A$$

Proof

Step	Hyp	Ref	Expression
1		df-ex	$$⊢\exists xx\in A\leftrightarrow \neg \forall x\neg x\in A$$
2		eq0	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$
3	1 2	xchbinxr	$$⊢\exists xx\in A\leftrightarrow \neg A=\varnothing$$
4	3	bicomi	$$⊢\neg A=\varnothing \leftrightarrow \exists xx\in A$$