Theorem suprcl

Description: Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Oct-2004)

	Ref	Expression
Assertion	suprcl	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\to sup\left(A,ℝ,<\right)\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		ltso	$$⊢<\mathrm{Or}ℝ$$
2	1	a1i	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\to <\mathrm{Or}ℝ$$
3		sup3	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\to \exists x\in ℝ\left(\forall y\in A\neg x<y\wedge \forall y\in ℝ\left(y<x\to \exists z\in Ay<z\right)\right)$$
4	2 3	supcl	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\to sup\left(A,ℝ,<\right)\in ℝ$$