I'm working on some set theory problems and I've run across some issues. I need to prove:
(Sorry if this looks messy but I dont know exactly how to type this out. It's a union of a collection of sets, by the way.) $\bigcup_{X\in\{A,B\}} X=A\cup B.$
So I start off using the definition of $\bigcup$ and I get:
$\forall x\colon(\exists X\colon X\in\{A,B\}\land x\in X)$
So my question is...can I go ahead and assume that $X$ is an element of $A \cup B$ since it is an element of $\{A,B\}$?
And then my next step would look like:
$(\forall X)(X \in A \cup B \Rightarrow x \in X)$