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Metamath Proof Explorer

Theorem elfz5

Description: Membership in a finite set of sequential integers. (Contributed by NM, 26-Dec-2005)

Ref Expression
Assertion elfz5 ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝐾 ≤ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 eluzelz ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝐾 ∈ ℤ )
2 eluzel2 ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
3 1 2 jca ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ) )
4 elfz ⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
5 4 3expa ⊢ ( ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
6 3 5 sylan ⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
7 eluzle ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ≤ 𝐾 )
8 7 biantrurd ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ≤ 𝑁 ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
9 8 adantr ⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ≤ 𝑁 ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
10 6 9 bitr4d ⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝐾 ≤ 𝑁 ) )