Theorem suprubii

Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by NM, 12-Sep-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	suprubii	$$⊢B\in A\to B\le sup\left(A,ℝ,<\right)$$

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