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Math Help - set theory problem

  1. #1
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    set theory problem

    How can we do this?
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  2. #2
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    x \in \left[ {\bigcup\limits_\alpha  {A_\alpha  } \backslash \bigcup\limits_\alpha  {B_\alpha  } } \right]\; \Rightarrow \;x \in \left[ {\bigcup\limits_\alpha  {A_\alpha  } } \right] \wedge x \notin \left[ {\bigcup\limits_\alpha  {B_\alpha  } } \right]
    x \in \left[ {\bigcup\limits_\alpha  {A_\alpha  } } \right] \wedge x \notin \left[ {\bigcup\limits_\alpha  {B_\alpha  } } \right]\; \Rightarrow \;\left( {\exists \gamma } \right)\left[ {x \in A_\gamma  } \right] \wedge \left( {\forall \alpha } \right)\left[ {x \notin B_\alpha  } \right]
    \begin{gathered}<br />
  \left( {\exists \gamma } \right)\left[ {x \in A_\gamma  } \right] \wedge \left( {\forall \alpha } \right)\left[ {x \notin B_\alpha  } \right]\; \Rightarrow \;x \in A_\gamma  \backslash B_\gamma   \hfill \\<br />
  x \in A_\gamma  \backslash B_\gamma  \; \Rightarrow \;x \in \left[ {\bigcup\limits_\alpha  {A_\alpha  \backslash B_\alpha  } } \right]\; \hfill \\ <br />
\end{gathered}
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