Singleton of Element is Subset/Proof 1

Theorem

Let $S$ be a set.

Let $\set x$ be the singleton of $x$.

Then:

$x \in S \iff \set x \subseteq S$

Proof

\(\ds \)

\(\)

\(\ds \set x \subseteq A\)

\(\ds \)

\(\leadstoandfrom\)

\(\ds \forall y: \paren {y \in \set x \implies y \in A}\)

Definition of Subset

\(\ds \)

\(\leadstoandfrom\)

\(\ds \forall y: \paren {y = x \implies y \in A}\)

Definition of Singleton

\(\ds \)

\(\leadstoandfrom\)

\(\ds x \in A\)

Equality implies Substitution

$\blacksquare$