Theorem elfzole1

Description: A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015)

	Ref	Expression
Assertion	elfzole1	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ≤ 𝐾 )

Proof

Step	Hyp	Ref	Expression
1		elfzoelz	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ℤ )
2		elfzoel1	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ℤ )
3		elfzoel2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
4		elfzo	⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁 ) ) )
6	5	ibi	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑀 ≤ 𝐾 ∧ 𝐾 < 𝑁 ) )
7	6	simpld	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ≤ 𝐾 )