Set Intersection Preserves Subsets/Corollary/Proof 2

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:

\(\ds \)

\(\)

\(\ds A \subseteq B\)

\(\ds \)

\(\leadsto\)

\(\ds \paren {x \in A \implies x \in B}\)

Definition of Subset

\(\ds \)

\(\leadsto\)

\(\ds \paren {x \in A \land x \in S \implies x \in B \land x \in S}\)

Factor Principles

\(\ds \)

\(\leadsto\)

\(\ds \paren {x \in A \cap S \implies x \in B \cap S}\)

Definition of Set Intersection

\(\ds \)

\(\leadsto\)

\(\ds A \cap S \subseteq B \cap S\)

Definition of Subset

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(e)}$