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No - let $\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega, n \geq 2\}$$\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega\}$.
No - let $\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega, n \geq 2\}$
No - let $\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega\}$.
