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 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-ne	⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ )
2		neq0	⊢ ( ¬ 𝐴 = ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
3	1 2	bitri	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )