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 ( 𝜑𝑀 ∈ ℤ )
uzidd2.2 𝑍 = ( ℤ𝑀 )
Assertion uzidd2 ( 𝜑𝑀𝑍 )

Proof

Step Hyp Ref Expression
1 uzidd2.1 ( 𝜑𝑀 ∈ ℤ )
2 uzidd2.2 𝑍 = ( ℤ𝑀 )
3 1 uzidd ( 𝜑𝑀 ∈ ( ℤ𝑀 ) )
4 3 2 eleqtrrdi ( 𝜑𝑀𝑍 )