Set Intersection Preserves Subsets/Corollary/Proof 2

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Corollary to Set Intersection Preserves Subsets

Let $A, B, S$ be sets.

Then:

$A \subseteq B \implies A \cap S \subseteq B \cap S$


Proof

Recall the Factor Principles, themselves a corollary of the Praeclarum Theorema:

$\paren {p \implies q} \vdash \paren {p \land r} \implies \paren {q \land r}$


This is applied as:

\(\displaystyle \) \(\) \(\displaystyle A \subseteq B\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \in A \implies x \in B}\) Definition of Subset
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \in A \land x \in S \implies x \in B \land x \in S}\) Factor Principles
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \in A \cap S \implies x \in B \cap S}\) Definition of Set Intersection
\(\displaystyle \) \(\leadsto\) \(\displaystyle A \cap S \subseteq B \cap S\) Definition of Subset

$\blacksquare$


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