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Metamath Proof Explorer

Theorem eluzd

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

Ref Expression
Hypotheses eluzd.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
eluzd.2 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
eluzd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
eluzd.4 ⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
Assertion eluzd ( 𝜑 → 𝑁 ∈ 𝑍 )

Proof

Step Hyp Ref Expression
1 eluzd.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2 eluzd.2 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3 eluzd.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4 eluzd.4 ⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
5 eluz2 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) )
6 2 3 4 5 syl3anbrc ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
7 6 1 eleqtrrdi ⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )