Not quite. I live in a republic. The majority wins elections but still don't get to totally trample on the minority, because we have a limited government.

Now, a union election isn't a government election, so that doesn't totally apply. But, for example, the majority has decided to pass legislation that makes my current state a right-to-work state, so a union can't require me to be a member, even if they win the election.

Of course, unions could campaign to change that law. But that's a different set of elections they'd have to win...