Theorem infrglb

Description: The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017) (Revised by AV, 15-Sep-2020)

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

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		simp1	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ax\le y\right)\to A\subseteq ℝ$$
5	2 3 4	infglbb	$$⊢\left(\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ax\le y\right)\wedge B\in ℝ\right)\to \left(sup\left(A,ℝ,<\right)<B\leftrightarrow \exists z\in Az<B\right)$$