[A] An example.
Construct the real numbers, for example some complicated thing using cuts.
Define addition, multiplication, ordering for the real numbers.
Prove the real numbers make up a complete ordered field.
Prove: any two complete ordered fields are isomorphic.
From then on, use only "complete ordered field", and never mention cuts again.

[B] Simpler
(1) An "ordered pair" is defined $\langle a,b \rangle = \{\{a\},\{a,b\}\}$.
(2) Prove $\langle a,b \rangle = \langle c,d\rangle$ if and only if $a=c$ and $b=d$.
(3) From now on, use only the "defining property" (2). After all, that is the motivation for (1) in the first place.

[C] Another
Prove the following are equivalent (in ZF): (1) the axiom of choice (2) the well ordering princple (3) Zorn's lemma (4) Tukey's lemma (5) the Hausdorff maximality principle.
After that, when you say "ZFC" use any one of these.
(From Brendan McKay's comment.)