Metamath Proof Explorer
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 ) |