Theorem lbinfcl

Description: If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005) (Revised by AV, 4-Sep-2020)

	Ref	Expression
Assertion	lbinfcl	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\right)\to sup\left(S,ℝ,<\right)\in S$$

Proof

Step	Hyp	Ref	Expression
1		lbinf	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\right)\to sup\left(S,ℝ,<\right)=\left(\iota x\in S|\forall y\in Sx\le y\right)$$
2		lbcl	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\right)\to \left(\iota x\in S|\forall y\in Sx\le y\right)\in S$$
3	1 2	eqeltrd	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\right)\to sup\left(S,ℝ,<\right)\in S$$