Theorem sup3ii

Description: A version of the completeness axiom for reals. (Contributed by NM, 23-Aug-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	sup3ii	$$⊢\exists x\in ℝ\left(\forall y\in A\neg x<y\wedge \forall y\in ℝ\left(y<x\to \exists z\in Ay<z\right)\right)$$

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		sup3	$$⊢\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\to \exists x\in ℝ\left(\forall y\in A\neg x<y\wedge \forall y\in ℝ\left(y<x\to \exists z\in Ay<z\right)\right)$$
3	1 2	ax-mp	$$⊢\exists x\in ℝ\left(\forall y\in A\neg x<y\wedge \forall y\in ℝ\left(y<x\to \exists z\in Ay<z\right)\right)$$