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Definition D76
Set intersection

Let $x$ be a D11: Set.
The intersection of $x$ is the D11: Set $$\cap x : = \{ z \mid \forall \, y \in x : z \in y \}$$

Let $X_j$ be a D11: Set for each $j \in J$.
The intersection of $\{ X_j \}_{j \in J}$ is the D11: Set $$\bigcap_{j \in J} X_j : = \{ x \mid \forall \, j \in J : x \in X_j \}$$
Results
 ▶ Binary intersection is a lower bound to each set in the intersection ▶ Binary set intersection is commutative ▶ Countable intersection is a lower bound to each set in the intersection ▶ Countable set intersection is invariant under bijective shifting of indices ▶ Finite intersection is a lower bound to each set in the intersection ▶ Finite set intersection is invariant under bijective shifting of indices ▶ Intersection is a lower bound to each set in the intersection ▶ Intersection is largest lower bound ▶ Isotonicity of set intersection ▶ Set intersection with empty set is empty