# Category:Well-Orderings

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This category contains results about Well-Orderings.
Definitions specific to this category can be found in Definitions/Well-Orderings.

The ordering $\preceq$ is a well-ordering on $S$ if and only if every non-empty subset of $S$ has a smallest element under $\preceq$:

$\forall T \subseteq S: \exists a \in T: \forall x \in T: a \preceq x$