Theorem lbinfle

Description: If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005) (Revised by AV, 4-Sep-2020)

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

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	1	3adant3	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\wedge A\in S\right)\to sup\left(S,ℝ,<\right)=\left(\iota x\in S|\forall y\in Sx\le y\right)$$
3		lble	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\wedge A\in S\right)\to \left(\iota x\in S|\forall y\in Sx\le y\right)\le A$$
4	2 3	eqbrtrd	$$⊢\left(S\subseteq ℝ\wedge \exists x\in S\forall y\in Sx\le y\wedge A\in S\right)\to sup\left(S,ℝ,<\right)\le A$$