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	⊢ ( ¬ 𝐴 = ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-ex	⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
2		eq0	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
3	1 2	xchbinxr	⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 ↔ ¬ 𝐴 = ∅ )
4	3	bicomi	⊢ ( ¬ 𝐴 = ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )