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Like Stefan mentions, under AD every ultrafilter is $\omega_1$-complete. Then the proof that the c.u.b filter on $\omega_1$ is an ultrafilter proof goes through Solovay's game where players play codes for well-orderings. The players choose countable ordinal ordinals $\alpha_i$ for $i < \omega$. Player I wins if $sup${$\alpha_i:i<\omega$} $\in Y \subset \omega_1$ for some $Y \subset \omega_1$ over which the game is played.
Like Stefan mentions, under AD every ultrafilter is $\omega_1$-complete. Then the proof that the c.u.b filter on $\omega_1$ is an ultrafilter proof goes through Solovay's game where players play codes for well-orderings. The players choose countable ordinal $\alpha_i$ for $i < \omega$. Player I wins if $sup${$\alpha_i:i<\omega$} $\in Y \subset \omega_1$ for some $Y \subset \omega_1$ over which the game is played.