The main result of the paper is an extension of the resolution proof procedure based on A-orderings of literals (i.e., strict partial orderings respecting substitutions) to the tableau proof procedure. The authors give a simple completeness proof for A-ordered ground clause tableaux and then extend this technique to non-ground non-clausal formulas in the negation normal form by proving completeness of an A-ordered tableau procedure. It is noted that this procedure is not compatible with the connection method.