Singleton Equality

This page has been identified as a candidate for refactoring of basic complexity.
In particular: Two results here.
Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Theorems

Let $x$ and $y$ be sets.

Then:

$\set x \subseteq \set y \iff x = y$

$\set x = \set y \iff x = y$

Proof

$\ds $

$\ds $

$\ds \set x \subseteq \set y$

$\ds $

$\leadstoandfrom$

$\ds \forall z:$

$\ds \paren {z \in \set x \implies z \in \set y}$

Definition of Subset

$\ds $

$\leadstoandfrom$

$\ds \forall z:$

$\ds \paren {z = x \implies z = y}$

Definition of Singleton

$\ds $

$\leadstoandfrom$

$\ds $

$\ds x = y$

Equality implies Substitution

$\Box$

Then:

$\ds x$

$=$

$\ds y$

$\ds \leadsto \ \ $

$\ds \set x$

$=$

$\ds \set y$

Substitutivity of Equality

$\ds \leadsto \ \ $

$\ds \set x$

$\subseteq$

$\ds \set y$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds y$

by the first part

$\blacksquare$

Sources

1963: Willard Van Orman Quine: Set Theory and Its Logic: $\S 7.7$