Example 4

Tag 03TY, permanent. Cite it as [Bourbaki, Tag 03TY].

Theory of Sets, chapter III, § 1. Order relations. Ordered sets

If A is a set which has at least two distinct elements, the set $\mathfrak{P}(A)$ (ordered by inclusion) is not totally ordered, for if $x\neq y$ , the subsets $\{x\}$ and $\{y\}$ are not comparable.

A totally ordered set is also totally ordered with respect to the opposite ordering; it is a lattice and a fortiori both left and right directed.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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