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Metamath Proof Explorer

Theorem n0f

Description: A class is nonempty if and only if it has at least one element. Proposition 5.17(1) of TakeutiZaring p. 20. This version of n0 requires only that x not be free in, rather than not occur in, A . (Contributed by NM, 17-Oct-2003)

Ref Expression
Hypothesis eq0f.1 ⊢ Ⅎ _ x A
Assertion n0f ⊢ A ≠ ∅ ↔ ∃ x x ∈ A

Proof

Step Hyp Ref Expression
1 eq0f.1 ⊢ Ⅎ _ x A
2 df-ne ⊢ A ≠ ∅ ↔ ¬ A = ∅
3 1 neq0f ⊢ ¬ A = ∅ ↔ ∃ x x ∈ A
4 2 3 bitri ⊢ A ≠ ∅ ↔ ∃ x x ∈ A