Proposition 2

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

Theory of Sets, chapter II, § 1. Collectivizing relations

$(x \subset y$ and $y \subset z) \Rightarrow (x \subset z)$ .

Adjoin the hypotheses $x \subset y$ , $y \subset z$ , and $u \in x$ . Then the relations

(u \in x) \Rightarrow (u \in y), \qquad (u \in y) \Rightarrow (u \in z),

are true, and therefore the relation $u \in z$ is true.

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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