Theorem suprleubii

Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 6-Sep-2014)

	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	suprleubii	$$⊢B\in ℝ\to \left(sup\left(A,ℝ,<\right)\le B\leftrightarrow \forall z\in Az\le B\right)$$

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		suprleub	$$⊢\left(\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\wedge B\in ℝ\right)\to \left(sup\left(A,ℝ,<\right)\le B\leftrightarrow \forall z\in Az\le B\right)$$
3	1 2	mpan	$$⊢B\in ℝ\to \left(sup\left(A,ℝ,<\right)\le B\leftrightarrow \forall z\in Az\le B\right)$$