Subset Relation is Ordering/Class Theory
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Theorem
Let $C$ be a class.
Then the subset relation $\subseteq$ is an ordering on $C$.
Proof
To establish that $\subseteq$ is an ordering, we need to show that it is reflexive, antisymmetric and transitive.
So, checking in turn each of the criteria for an ordering:
Reflexivity
From Subset Relation is Reflexive:
- $\forall x \in C: x \subseteq x$
So $\subseteq$ is reflexive.
$\Box$
Antisymmetry
From Subset Relation is Antisymmetric:
- $\forall x, y \in C: x \subseteq y \land y \subseteq x \iff x = y$
So $\subseteq$ is antisymmetric.
$\Box$
Transitivity
From Subset Relation is Transitive:
- $\forall x, y, z \in C: x \subseteq y \land y \subseteq z \implies x \subseteq z$
That is, $\subseteq$ is transitive.
$\Box$
So we have shown that $\subseteq$ is an ordering on $C$.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering