Theorem iooss2

Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 3-Nov-2013)

	Ref	Expression
Assertion	iooss2	$$⊢\left(C\in {ℝ}^{*}\wedge B\le C\right)\to \left(A,B\right)\subseteq \left(A,C\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-ioo	$$⊢\left(.\right)=\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼\left\{z\in {ℝ}^{*}|\left(x<z\wedge z<y\right)\right\}\right)$$
2		xrltletr	$$⊢\left(w\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to \left(\left(w<B\wedge B\le C\right)\to w<C\right)$$
3	1 1 2	ixxss2	$$⊢\left(C\in {ℝ}^{*}\wedge B\le C\right)\to \left(A,B\right)\subseteq \left(A,C\right)$$