Theorem elfzolt2b

Description: A member in a half-open integer interval is less than the upper bound. (Contributed by Mario Carneiro, 29-Sep-2015)

	Ref	Expression
Assertion	elfzolt2b	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ( 𝐾 ..^ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzoelz	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ℤ )
2		elfzoel2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
3		elfzolt2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 < 𝑁 )
4		fzolb	⊢ ( 𝐾 ∈ ( 𝐾 ..^ 𝑁 ) ↔ ( 𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 < 𝑁 ) )
5	1 2 3 4	syl3anbrc	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ( 𝐾 ..^ 𝑁 ) )