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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 ≠ ∅