Editing Axiom of regularity (section)

=== Every set has an ordinal rank ===
This was actually the original form of the axiom in von Neumann's axiomatization.

Suppose ''x'' is any set. Let ''t'' be the [[transitive closure (set)|transitive closure]] of {''x''}. Let ''u'' be the subset of ''t'' consisting of unranked sets. If ''u'' is empty, then ''x'' is ranked and we are done. Otherwise, apply the axiom of regularity to ''u'' to get an element ''w'' of ''u'' which is disjoint from ''u''. Since ''w'' is in ''u'', ''w'' is unranked. ''w'' is a subset of ''t'' by the definition of transitive closure. Since ''w'' is disjoint from ''u'', every element of ''w'' is ranked. Applying the axioms of replacement and union to combine the ranks of the elements of ''w'', we get an ordinal rank for ''w'', to wit <math display="inline">\textstyle \operatorname{rank} (w) = \cup \{ \operatorname{rank} (z) + 1 \mid z \in w \}</math>. This contradicts the conclusion that ''w'' is unranked. So the assumption that ''u'' was non-empty must be false and ''x'' must have rank.