Theorem elicod

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

	Ref	Expression
Hypotheses	elicod.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	elicod.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	elicod.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
	elicod.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
	elicod.5	⊢ ( 𝜑 → 𝐶 < 𝐵 )
Assertion	elicod	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 [,) 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elicod.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		elicod.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		elicod.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		elicod.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )
5		elicod.5	⊢ ( 𝜑 → 𝐶 < 𝐵 )
6		elico1	⊢ ( ( 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 ) )