I wrote a little "math syntax" to "english words" dictionary which might help you: http://mcgillweb.alwaysdata.net/notation_appendix.pdf

In particular the section on set theory is important (element of, subset, etc..) and the two qualifiers: for all, and there exists. Here is a 6 page primer on set notation, which contains a simple proof http://mcgillweb.alwaysdata.net/set_notation.pdf