Equality is Reflexive

Theorem

Equality is reflexive.

That is:

$\forall a: a = a$

Proof

This proof depends on Leibniz's law:

$x = y \dashv \vdash \map P x \iff \map P y$

We are trying to prove $a = a$.

Our assertion, then, is:

$a = a \dashv \vdash \map P a \iff \map P a$

From Law of Identity, $\map P a \iff \map P a$ is a tautology.

Thus $a = a$ is also tautologous, and the theorem holds.

$\blacksquare$

Also see

• Equality is Symmetric
• Equality is Transitive
• Equality is Equivalence Relation

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.): Chapter $3$
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 1$: The Axiom of Extension
• 1971: Wilfred Kaplan and Donald J. Lewis: Calculus and Linear Algebra ... (previous) ... (next): Introduction: Review of Algebra, Geometry, and Trigonometry: $\text{0-1}$: The Real Numbers
• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 3.2$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Equality: $\text{(a)}$