Theorem eliccelicod

Description: A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	eliccelicod.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	eliccelicod.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	eliccelicod.c	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
	eliccelicod.d	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
Assertion	eliccelicod	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eliccelicod.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		eliccelicod.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		eliccelicod.c	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) )
4		eliccelicod.d	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
5		eliccxr	⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → 𝐶 ∈ ℝ* )
6	3 5	syl	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
7		iccgelb	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝐶 )
8	1 2 3 7	syl3anc	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
9		iccleub	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ≤ 𝐵 )
10	1 2 3 9	syl3anc	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )
11	6 2 10 4	xrleneltd	⊢ ( 𝜑 → 𝐶 < 𝐵 )
12	1 2 6 8 11	elicod	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,) 𝐵 ) )