Theorem neq0f

Description: A class is not empty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. This version of neq0 requires only that x not be free in, rather than not occur in, A . (Contributed by BJ, 15-Jul-2021)

	Ref	Expression
Hypothesis	eq0f.1	⊢ Ⅎ 𝑥 𝐴
Assertion	neq0f	⊢ ( ¬ 𝐴 = ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		eq0f.1	⊢ Ⅎ 𝑥 𝐴
2	1	eq0f	⊢ ( 𝐴 = ∅ ↔ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
3	2	notbii	⊢ ( ¬ 𝐴 = ∅ ↔ ¬ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
4		df-ex	⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 ↔ ¬ ∀ 𝑥 ¬ 𝑥 ∈ 𝐴 )
5	3 4	bitr4i	⊢ ( ¬ 𝐴 = ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )