The axiom of Zermelo-Fraenkel set theory which asserts that sets formed by the same elements are equal, for all x(x element a congruent x element b)⇒a = b. Note that some texts, use a bidirectional equivalent congruent preceding "a = b, " while others (e.g., Enderton 1977, Itô 1986), use the one-way implies ⇒. However, one-way implication suffices. Using the notation a subset b (a is a subset of b) for (x element a)⇒(x element b), the axiom can be written concisely as a subset b⋀b subset a⇒a = b, where ⋀ denotes logical AND.
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