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Metamath Proof Explorer

Theorem elfzod

Description: Membership in a half-open integer interval. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses elfzod.1 ⊢ φ → K ∈ ℤ ≥ M
elfzod.2 ⊢ φ → N ∈ ℤ
elfzod.3 ⊢ φ → K < N
Assertion elfzod ⊢ φ → K ∈ M ..^ N

Proof

Step Hyp Ref Expression
1 elfzod.1 ⊢ φ → K ∈ ℤ ≥ M
2 elfzod.2 ⊢ φ → N ∈ ℤ
3 elfzod.3 ⊢ φ → K < N
4 elfzo2 ⊢ K ∈ M ..^ N ↔ K ∈ ℤ ≥ M ∧ N ∈ ℤ ∧ K < N
5 1 2 3 4 syl3anbrc ⊢ φ → K ∈ M ..^ N