Theorem elioc1

Description: Membership in an open-below, closed-above interval of extended reals. (Contributed by NM, 24-Dec-2006) (Revised by Mario Carneiro, 3-Nov-2013)

	Ref	Expression
Assertion	elioc1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(C\in \left(A,B\right]\leftrightarrow \left(C\in {ℝ}^{*}\wedge A<C\wedge C\le B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ioc	$$⊢\left(.\right]=\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼\left\{z\in {ℝ}^{*}|\left(x<z\wedge z\le y\right)\right\}\right)$$
2	1	elixx1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(C\in \left(A,B\right]\leftrightarrow \left(C\in {ℝ}^{*}\wedge A<C\wedge C\le B\right)\right)$$