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Metamath Proof Explorer

Theorem uzn0d

Description: The upper integers are all nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses uzn0d.1 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
uzn0d.2 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
Assertion uzn0d ( 𝜑 → 𝑍 ≠ ∅ )

Proof

Step Hyp Ref Expression
1 uzn0d.1 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2 uzn0d.2 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3 1 2 uzidd2 ⊢ ( 𝜑 → 𝑀 ∈ 𝑍 )
4 3 ne0d ⊢ ( 𝜑 → 𝑍 ≠ ∅ )