Can someone verify/correct this proof?

Can you conclude that A=B if A,B, and C are sets such that:

A U C = B U C and A intersection C = B intersection C

I know its true but I'm not sure of the proof:

pf: Let A,B,C be sets.

Suppose x be in AUC. Then x is in BUC.

Then x is in A and x is in B or x is in.

Suppose x is in A intersect C. Then x is in B intersect C.

Then x is A, x is in B, and x is in C.

Since x is in A and x is in then x is in AUC, BUC.

Since x is in C, then x is in AUC, BUC.

Thus, from these last two lines, x in A intersect C and x is in B intersect C.

Therefore, if x is in AUC=BUC then x is automatically in A and B or in C.

If x is in A intersect C = B intersect C, then x is A, B, and C.

A=B since x is in the intersection of A and C and intersection of B and C and also since x appears in the AUC and BUC regardless of C, because of the statement x is in A and x is in B or x is in C.

I really don't think this proves anything...anyone know of a more obvious or quick approach?