Simple Proof of Obvious Fact. Have you any idea of how to proof the following as simpler as possible?
Let $M$ be family of $n$ finite sets:
$$M = \{X_1, X_2, ..., X_n\}$$
Then 
$$\bigr(\forall k = 1, 2, ..., n : |X_{i_{1}}\cup X_{i_{2}}\cup\cdots\cup X_{i_{k}}|\geq k\bigl)\Rightarrow\bigl(\exists \chi_1,\chi_2,...\chi_n : \chi_i\neq\chi_j,\,\chi_i\in X_i\bigr)$$
In other words
If union of any k sets has at least k elements then there exist n different elements which represent whole family M.

Your obvious fact is a harder part of a generalization of Hall’s Marriage Theorem, proved in 1935 and it probably has no very simple proof (at least I was not told about it in my graph theory lecture course). 
