Theorem n0

Description: A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. (Contributed by NM, 29-Sep-2006) Avoid ax-11 , ax-12 . (Revised by Gino Giotto, 28-Jun-2024)

	Ref	Expression
Assertion	n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$

Proof

Step	Hyp	Ref	Expression
1		df-ne	$$⊢A\ne \varnothing \leftrightarrow \neg A=\varnothing$$
2		neq0	$$⊢\neg A=\varnothing \leftrightarrow \exists xx\in A$$
3	1 2	bitri	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$