Theorem suprnub

Description: An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by NM, 15-Nov-2004) (Revised by Mario Carneiro, 6-Sep-2014)

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

Proof

Step	Hyp	Ref	Expression
1		suprlub	$$⊢\left(\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\wedge B\in ℝ\right)\to \left(B<sup\left(A,ℝ,<\right)\leftrightarrow \exists z\in AB<z\right)$$
2	1	notbid	$$⊢\left(\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\wedge B\in ℝ\right)\to \left(\neg B<sup\left(A,ℝ,<\right)\leftrightarrow \neg \exists z\in AB<z\right)$$
3		ralnex	$$⊢\forall z\in A\neg B<z\leftrightarrow \neg \exists z\in AB<z$$
4	2 3	bitr4di	$$⊢\left(\left(A\subseteq ℝ\wedge A\ne \varnothing \wedge \exists x\in ℝ\forall y\in Ay\le x\right)\wedge B\in ℝ\right)\to \left(\neg B<sup\left(A,ℝ,<\right)\leftrightarrow \forall z\in A\neg B<z\right)$$