This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 GG, 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