Proving that $\langle\mathbb{R}-\{7\},<\rangle$ is an elementary substructure of $\langle\mathbb{R},<\rangle$

Question

Prove that $\langle\mathbb{R}-\{7\},<\rangle$ is an elementary substructure of $\langle\mathbb{R},<\rangle$.

What I thought I should do is to use induction on statements to prove that. Any easier way to acomplish that?

Answer 1

An easier way is an application of the following "sufficient condition"-style test.

If $\mathfrak{A}=\langle A,\ldots\rangle$ is a substructure of $\mathfrak{B}=\langle B,\ldots\rangle$ and, for any finite subset $A'$ of $A$ and any $b\in B$, there is an automorphism $f$ of $\mathfrak{B}$ such that $f(a)=a$ for any $a\in A'$, and $f(b)\in A$, then $\mathfrak{A}$ is an elementary substructure of $\mathfrak{B}$.

(Which may appear in your prerequisite course, maybe in somewhat stronger form, and is usually proven using Tarski–Vaught criterion). It is easy to exhibit an $f$ in your setup.