Theorem suprclii

Description: Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Sep-1999)

	Ref	Expression
Hypothesis	sup3i.1	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)$$
Assertion	suprclii	$$⊢sup\left(A,ℝ,<\right)\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		sup3i.1	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)$$
2		suprcl	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\to sup\left(A,ℝ,<\right)\in ℝ$$
3	1 2	ax-mp	$$⊢sup\left(A,ℝ,<\right)\in ℝ$$