Equality is Reflexive
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Theorem
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
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: $\text{(a)}$