Uniqueness of $\mathbb{R}$ In the very beginning of baby Rudin we are given 

$\textbf{1.19 Theorem}$ There exists an ordered field $\textit{R}$ with the least upper bound axiom. Moreover, $\textit{R}$ contains $\textit{Q}$ as a subfield.

Are the real numbers the $\textit{unique}$ ordered field with l.u.b. property with the rationals being a subfield? Of course we know $\mathbb{R}$ exists through many constructions, but are there any uncountably infinite fields with the same properties that aren't $\mathbb{R}$ up to isomorphism?
My intuition is telling me that it's true since the fact that $\mathbb{Q}$ (or an isomorphic copy) being a subfield of each must mean that both fields must capture the same properties. But I'm really not sure. Apologies if this question is silly. 

Yes, the real numbers are unique in the sense you described. See the epilogue of Spivak's Calculus (chapter 29 or 30, depending on the edition).
