Singleton Equality

Theorems

Let $x$ and $y$ be sets.

Then:

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

Proof

 $\displaystyle$  $\displaystyle$ $\displaystyle \set x \subseteq \set y$ $\displaystyle$ $\leadstoandfrom$ $\displaystyle \forall z:$ $\displaystyle \paren {z \in \set x \implies z \in \set y}$ Definition of Subset $\displaystyle$ $\leadstoandfrom$ $\displaystyle \forall z:$ $\displaystyle \paren {z = x \implies z = y}$ Definition of Singleton $\displaystyle$ $\leadstoandfrom$ $\displaystyle$ $\displaystyle x = y$ Equality implies Substitution

$\Box$

Then:

 $\displaystyle x$ $=$ $\displaystyle y$ $\displaystyle \leadsto \ \$ $\displaystyle \set x$ $=$ $\displaystyle \set y$ Substitutivity of Equality $\displaystyle \leadsto \ \$ $\displaystyle \set x$ $\subseteq$ $\displaystyle \set y$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y$ by the first part

$\blacksquare$