Theorem fzofi

Description: Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015)

	Ref	Expression
Assertion	fzofi	⊢ ( 𝑀 ..^ 𝑁 ) ∈ Fin

Proof

Step	Hyp	Ref	Expression
1		fzoval	⊢ ( 𝑁 ∈ ℤ → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
2	1	adantl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
3		fzfi	⊢ ( 𝑀 ... ( 𝑁 − 1 ) ) ∈ Fin
4	2 3	eqeltrdi	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ..^ 𝑁 ) ∈ Fin )
5		fzof	⊢ ..^ : ( ℤ × ℤ ) ⟶ 𝒫 ℤ
6	5	fdmi	⊢ dom ..^ = ( ℤ × ℤ )
7	6	ndmov	⊢ ( ¬ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ..^ 𝑁 ) = ∅ )
8		0fi	⊢ ∅ ∈ Fin
9	7 8	eqeltrdi	⊢ ( ¬ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ..^ 𝑁 ) ∈ Fin )
10	4 9	pm2.61i	⊢ ( 𝑀 ..^ 𝑁 ) ∈ Fin