Theorem elfz1

Description: Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005)

	Ref	Expression
Assertion	elfz1	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzval	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ... 𝑁 ) = { 𝑗 ∈ ℤ ∣ ( 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁 ) } )
2	1	eleq2d	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝐾 ∈ { 𝑗 ∈ ℤ ∣ ( 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁 ) } ) )
3		breq2	⊢ ( 𝑗 = 𝐾 → ( 𝑀 ≤ 𝑗 ↔ 𝑀 ≤ 𝐾 ) )
4		breq1	⊢ ( 𝑗 = 𝐾 → ( 𝑗 ≤ 𝑁 ↔ 𝐾 ≤ 𝑁 ) )
5	3 4	anbi12d	⊢ ( 𝑗 = 𝐾 → ( ( 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁 ) ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
6	5	elrab	⊢ ( 𝐾 ∈ { 𝑗 ∈ ℤ ∣ ( 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁 ) } ↔ ( 𝐾 ∈ ℤ ∧ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
7		3anass	⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ↔ ( 𝐾 ∈ ℤ ∧ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
8	6 7	bitr4i	⊢ ( 𝐾 ∈ { 𝑗 ∈ ℤ ∣ ( 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁 ) } ↔ ( 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) )
9	2 8	syl6bb	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝐾 ∈ ℤ ∧ 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )