If $A$, $B$ are path connected and $A \cup B$ is simply connected, $A \cap B$ is path connected The only proof I know involves the Mayer - Vietoris sequence. Is there an elementary proof?
 A: Let $x, y \in A \cap B$. Let $\alpha : [0, 1] \rightarrow A$ be a path from $x$ to $y$, and let $\beta : [0, 1] \rightarrow B$ be a path from $x$ to $y$. Let $H : [0, 1] \times [0, 1] \rightarrow A \cup B$ be a homotopy of $\alpha$ to $\beta$ with fixed endpoints. Divide $[0, 1] \times [0, 1]$ into subsquares so tiny that each subsquare gets mapped by $H$ entirely to $A$ or to $B$ (i.e. consider the covering of the square given by $\{H^{-1}(A), H^{-1}(B)\}$ and take its Lebesgue number $\epsilon > 0$, then divide the square $[0, 1] \times [0, 1]$ into subsquares of diagonal $< \epsilon$). Now color each of the subsquares blue if gets mapped entirely to $A$, red if it gets mapped entirely to $B$, and black if it gets mapped entirely to $A \cap B$. It suffices to convince yourself that, starting at the left side of the square, you can travel to the right side of the square, passing only through edges that bound either a black square or two squares, one being red and the other being blue. Since endpoints are fixed in the homotopy, this will give you a path from $x$ to $y$ that lies in $A \cap B$.
A: As Qiacho comments, the result does follow from the groupoid version of the Seifert-van Kampen Theorem, which dates from 1967,  is exploited in the book Topology and Groupoids, and discussed on this mathoverview question and answer. This paper gives a relevant small correction to one part of the book. 
In effect, Pedro is running through part of the proof of that theorem. 
