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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 )