Simple Cartesian Product Proof(s)

[Heute]

Homework Statement

Prove for all sets A, B, and C that:
Ax(B^C) = (AxB)^(AxC)

Homework Equations

AxB is the Cartesian Product of A and B. That is, the set of all ordered pairs (x,y) such that x is an element of A and y is an element of B.
^ denotes intersection (and)

The Attempt at a Solution

(x,y) is an element of Ax(B^C)
iff x is an element of A and y is an element of (B^C)
iff x is an element of A and y is an element of B and y is an element of C
iff (x is an element of A and y is an element of B) and (x is an element of A and y is an element of C)
iff (x,y) is an element of (AxB) and (x,y) is an element of (AxC)
iff (x,y) is an element of (AxB)^(AxC)

I'm confident that I'm "right," but I did sort of produce a second "x is an element of A" out of nowhere to tidy up my sequence of logic. Is this poor form? What would be better?

 

[grey_earl]

This is alright, and not at all "poor form". If $$x \in A$$, then also $$x \in A \wedge x \in A$$.