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Metamath Proof Explorer

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