Metamath Proof Explorer

Theorem elfz4

Description: Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfz4 ⊢ M ∈ ℤ ∧ N ∈ ℤ ∧ K ∈ ℤ ∧ M ≤ K ∧ K ≤ N → K ∈ M … N

Proof

Step Hyp Ref Expression
1 elfz2 ⊢ K ∈ M … N ↔ M ∈ ℤ ∧ N ∈ ℤ ∧ K ∈ ℤ ∧ M ≤ K ∧ K ≤ N
2 1 biimpri ⊢ M ∈ ℤ ∧ N ∈ ℤ ∧ K ∈ ℤ ∧ M ≤ K ∧ K ≤ N → K ∈ M … N