Equivalent Definitions of Synthetic Basis

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Theorem

Let $X$ be a set.


Then for a subset $\mathcal B \subseteq \mathcal P \left({X}\right)$ of the power set of $X$, the following are equivalent:

$(1): \quad$ $\displaystyle \forall A, B \in \mathcal B: \exists \mathcal A \subseteq \mathcal B: A \cap B = \bigcup \mathcal A$.
$(2): \quad$ $\displaystyle \forall A, B \in \mathcal B: \forall x \in A \cap B: \exists W \in \mathcal B: x \in W \subseteq A \cap B$.


Proof

$(1)$ implies $(2)$

Suppose that $A, B \in \mathcal B$.

By hypothesis:

$\displaystyle \exists \mathcal A \subseteq \mathcal B: A \cap B = \bigcup \mathcal A$

By Set is Subset of Union: General Result, we have:

$\displaystyle \forall W \in \mathcal A: W \subseteq \bigcup \mathcal A = A \cap B$

Therefore:

$\displaystyle \forall x \in A \cap B: \exists W \in \mathcal A \subseteq \mathcal B: x \in W \subseteq A \cap B$

$\Box$


$(2)$ implies $(1)$

Suppose that $A, B \in \mathcal B$.

Define the set:

$\displaystyle \mathcal A = \left\{{W \in \mathcal B: W \subseteq A \cap B}\right\} \subseteq \mathcal B$


By Union is Smallest Superset: General Result, we have:

$\displaystyle \bigcup \mathcal A \subseteq A \cap B$


By hypothesis, we have:

$\displaystyle \forall x \in A \cap B: \exists W \in \mathcal A: x \in W$

Hence, by the definition of a subset and the definition of set union:

$\displaystyle A \cap B \subseteq \bigcup \mathcal A$


Therefore, by definition of set equality:

$\displaystyle A \cap B = \bigcup \mathcal A$

$\blacksquare$


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