Why do we even need transitivity? I am referring to two Peano axioms: If $x = y$ then $y = x$ (symmetry) If $x = y$ and $y = z$ then $x = z$ (transitivity) What I don't understand is why we need the transitive axiom. Isn't it already implied by the symmetry axiom? Is it more of a convenience? Can we prove that transitivity is a required axiom? What happens if we were to remove it? Or should I think of this more like "symmetry just says we can physically flip the order of the equalities, and transitivity lets us swap things around as long as they're all equal to each other"?

Consider the following relation: $x$ **opposes** $y$ if and only if $x = -y$.

Can you see that the "opposes" relation is symmetrical but not transitive?