Consider a card game where the deck consists of 63 distinct cards. The deck is created in the following manner: each card consists of some number of symbols, where no two symbols are the same. There are six different symbols. We have $\binom{6}{1}=6$ cards with 1 symbol each on them, we have $\binom{6}{2}=15$ cards with 2 symbols each on them, and so on until we reach the single $\binom{6}{6}=1$ card that has all six symbols.
A set consists of some number of card such that within the set, the number of times each symbol appears is even.
How can we prove that given any 7 cards within this deck, we can find at least one set?