= Nonstandard prime in an elementary extension of the integers
Name every <integer> and add a new constant asserted to be prime and unequal to every named <integer>. Every finite part is satisfied by an ordinary prime outside a finite list, using <Euclid's theorem>. <Compactness theorem> with the <elementary diagram> gives an <elementary extension> with the desired prime. A <countable> <elementary substructure> retaining that prime cannot be <isomorphic> to the <integer> <ring>, since any unital <ring isomorphism> fixes every standard <integer>.
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