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Metamath Proof Explorer

Theorem uzidd2

Description: Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses uzidd2.1 ⊢ φ → M ∈ ℤ
uzidd2.2 ⊢ Z = ℤ ≥ M
Assertion uzidd2 ⊢ φ → M ∈ Z

Proof

Step Hyp Ref Expression
1 uzidd2.1 ⊢ φ → M ∈ ℤ
2 uzidd2.2 ⊢ Z = ℤ ≥ M
3 1 uzidd ⊢ φ → M ∈ ℤ ≥ M
4 3 2 eleqtrrdi ⊢ φ → M ∈ Z