Equals is an Equivalence Relation
From ProofWiki
Theorem
Equals is an equivalence relation.
Proof
This follows from the axioms of equality:
- Equality is Reflexive: $\forall a: a = a$.
- Equality is Symmetric: $\forall a, b: a = b \implies b = a$.
- Equality is Transitive: $\forall a, b, c: a = b \land b = c \implies a = c$.
$\blacksquare$
Sources
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.2$: Example $29$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 16$
