Theorem iocleub

Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	iocleub	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐶 ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elioc1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) )
2		simp3	⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵 ) → 𝐶 ≤ 𝐵 )
3	1 2	biimtrdi	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,] 𝐵 ) → 𝐶 ≤ 𝐵 ) )
4	3	3impia	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐶 ≤ 𝐵 )