Theorem elfzolt3

Description: Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Stefan O'Rear, 15-Aug-2015)

	Ref	Expression
Assertion	elfzolt3	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 < 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		elfzoel1	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ℤ )
2	1	zred	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ℝ )
3		elfzoelz	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ℤ )
4	3	zred	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ℝ )
5		elfzoel2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
6	5	zred	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℝ )
7		elfzole1	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ≤ 𝐾 )
8		elfzolt2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 < 𝑁 )
9	2 4 6 7 8	lelttrd	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 < 𝑁 )