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Math Help - More Axiom of Union and Powerset

  1. #1
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    More Axiom of Union and Powerset

    Proof.
    Last edited by abc512; August 29th 2009 at 12:19 PM. Reason: clarity.
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  2. #2
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    Let y \in x and let z \in y. Then z \in \cup x (definition of union). So y \subseteq \cup x. It follows that y \in \mathcal{P} \left(\cup x \right) (definition of power set).
    We showed that y \in x implies y \in \mathcal{P} \left(\cup x \right), so x \subseteq \mathcal{P} \left(\cup x \right).

    Example of strict inclusion: x = \{\{a\}\}. We have \cup x = \{a\}, \mathcal{P}\left(\cup x \right) = \{\emptyset, \{a\}\}. We see that x = \{\{a\}\} \subsetneq \{\emptyset, \{a\}\} = \mathcal{P}\left(\cup x \right).
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