Theorem elicc01

Description: Membership in the closed real interval between 0 and 1, also called the closed unit interval. (Contributed by AV, 20-Aug-2022)

	Ref	Expression
Assertion	elicc01	⊢ ( 𝑋 ∈ ( 0 [,] 1 ) ↔ ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1 ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		1re	⊢ 1 ∈ ℝ
3	1 2	elicc2i	⊢ ( 𝑋 ∈ ( 0 [,] 1 ) ↔ ( 𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1 ) )