Theorem eliocd

Description: Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	eliocd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	eliocd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	eliocd.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
	eliocd.altc	⊢ ( 𝜑 → 𝐴 < 𝐶 )
	eliocd.cleb	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )
Assertion	eliocd	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eliocd.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		eliocd.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		eliocd.c	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		eliocd.altc	⊢ ( 𝜑 → 𝐴 < 𝐶 )
5		eliocd.cleb	⊢ ( 𝜑 → 𝐶 ≤ 𝐵 )
6		elioc1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
7	1 2 6	syl2anc	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
8	3 4 5 7	mpbir3and	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 (,] 𝐵 ) )