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Metamath Proof Explorer

Theorem icogelbd

Description: An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	icogelbd.1	$⊢ φ \to A \in ℝ^{*}$
	icogelbd.2	$⊢ φ \to B \in ℝ^{*}$
	icogelbd.3	$⊢ φ \to C \in [A, B)$
Assertion	icogelbd	$⊢ φ \to A \leq C$

Proof

Step	Hyp	Ref	Expression
1		icogelbd.1	$⊢ φ \to A \in ℝ^{*}$
2		icogelbd.2	$⊢ φ \to B \in ℝ^{*}$
3		icogelbd.3	$⊢ φ \to C \in [A, B)$
4		icogelb	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land C \in [A, B)) \to A \leq C$
5	1 2 3 4	syl3anc	$⊢ φ \to A \leq C$