Theorem hashgt0n0

Description: If the size of a set is greater than 0, the set is not empty. (Contributed by AV, 5-Aug-2018) (Proof shortened by AV, 18-Nov-2018)

	Ref	Expression
Assertion	hashgt0n0	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 0 < ( ♯ ‘ 𝐴 ) ) → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		hashneq0	⊢ ( 𝐴 ∈ 𝑉 → ( 0 < ( ♯ ‘ 𝐴 ) ↔ 𝐴 ≠ ∅ ) )
2	1	biimpa	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 0 < ( ♯ ‘ 𝐴 ) ) → 𝐴 ≠ ∅ )