Metamath Proof Explorer

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	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in (A, B]) \to C \leq B$

Proof

Step	Hyp	Ref	Expression
1		elioc1	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in (A, B] \leftrightarrow (C \in ℝ^{*} \land A < C \land C \leq B))$
2		simp3	$⊢ (C \in ℝ^{*} \land A < C \land C \leq B) \to C \leq B$
3	1 2	syl6bi	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in (A, B] \to C \leq B)$
4	3	3impia	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in (A, B]) \to C \leq B$