# Metamath Proof Explorer

## Theorem suprleub

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
Assertion suprleub ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to \left(sup\left({A},ℝ,<\right)\le {B}↔\forall {z}\in {A}\phantom{\rule{.4em}{0ex}}{z}\le {B}\right)$

### Proof

Step Hyp Ref Expression
1 suprnub ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to \left(¬{B}
2 suprcl ${⊢}\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\to sup\left({A},ℝ,<\right)\in ℝ$
3 lenlt ${⊢}\left(sup\left({A},ℝ,<\right)\in ℝ\wedge {B}\in ℝ\right)\to \left(sup\left({A},ℝ,<\right)\le {B}↔¬{B}
4 2 3 sylan ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to \left(sup\left({A},ℝ,<\right)\le {B}↔¬{B}
5 simpl1 ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to {A}\subseteq ℝ$
6 5 sselda ${⊢}\left(\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\wedge {w}\in {A}\right)\to {w}\in ℝ$
7 simplr ${⊢}\left(\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\wedge {w}\in {A}\right)\to {B}\in ℝ$
8 6 7 lenltd ${⊢}\left(\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\wedge {w}\in {A}\right)\to \left({w}\le {B}↔¬{B}<{w}\right)$
9 8 ralbidva ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to \left(\forall {w}\in {A}\phantom{\rule{.4em}{0ex}}{w}\le {B}↔\forall {w}\in {A}\phantom{\rule{.4em}{0ex}}¬{B}<{w}\right)$
10 1 4 9 3bitr4d ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to \left(sup\left({A},ℝ,<\right)\le {B}↔\forall {w}\in {A}\phantom{\rule{.4em}{0ex}}{w}\le {B}\right)$
11 breq1 ${⊢}{w}={z}\to \left({w}\le {B}↔{z}\le {B}\right)$
12 11 cbvralvw ${⊢}\forall {w}\in {A}\phantom{\rule{.4em}{0ex}}{w}\le {B}↔\forall {z}\in {A}\phantom{\rule{.4em}{0ex}}{z}\le {B}$
13 10 12 syl6bb ${⊢}\left(\left({A}\subseteq ℝ\wedge {A}\ne \varnothing \wedge \exists {x}\in ℝ\phantom{\rule{.4em}{0ex}}\forall {y}\in {A}\phantom{\rule{.4em}{0ex}}{y}\le {x}\right)\wedge {B}\in ℝ\right)\to \left(sup\left({A},ℝ,<\right)\le {B}↔\forall {z}\in {A}\phantom{\rule{.4em}{0ex}}{z}\le {B}\right)$