iccleub

Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009)

		Ref	Expression
	Assertion	iccleub	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in [A, B]) \to C \leq B$

Step	Hyp	Ref	Expression
1		elicc1	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in [A, B] \leftrightarrow (C \in ℝ^{*} \land A \leq C \land C \leq B))$
2		simp3	$⊢ (C \in ℝ^{*} \land A \leq C \land C \leq B) \to C \leq B$
3	1 2	syl6bi	$⊢ (A \in ℝ^{} \land B \in ℝ^{}) \to (C \in [A, B] \to C \leq B)$
4	3	3impia	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in [A, B]) \to C \leq B$

Metamath Proof Explorer