Theorem infrecl

Description: Closure of infimum of a nonempty bounded set of reals. (Contributed by NM, 8-Oct-2005) (Revised by AV, 4-Sep-2020)

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

Proof

Step	Hyp	Ref	Expression
1		ltso	$$⊢<\mathrm{Or}ℝ$$
2	1	a1i	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ax\le y\right)\to <\mathrm{Or}ℝ$$
3		infm3	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ax\le y\right)\to \exists x\in ℝ\left(\forall y\in A\neg y<x\wedge \forall y\in ℝ\left(x<y\to \exists z\in Az<y\right)\right)$$
4	2 3	infcl	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ax\le y\right)\to sup\left(A,ℝ,<\right)\in ℝ$$