11Equivalent inclusion criteria

Before we go on to the operations on sets, here is a second general method for showing that two sets are identical. Call two conditions equivalent when every object satisfying one of them also satisfies the other:

Definition 7 (Equivalent conditions). Two conditions are equivalent when every object satisfying either one of them also satisfies the other.

Proposition 7 (equivalent criteria give the same set). If two sets have equivalent inclusion criteria, then they are identical.

Let X and Y be sets whose inclusion criteria are equivalent, and let x be any object. Then x ∈ X if and only if x satisfies the inclusion criterion for X, and likewise x ∈ Y if and only if x satisfies the inclusion criterion for Y. Since the two criteria are equivalent, x satisfies the one if and only if it satisfies the other. So x ∈ X if and only if x ∈ Y. As this holds for any object x, the two sets have exactly the same elements, and so they are identical by the Axiom of Extensionality.

The method this gives us is as follows: to show that two sets are identical, write down the inclusion criteria for each of them, and show that the two criteria are equivalent. We will not cite Proposition 7 (equivalent criteria give the same set) when we use the method, but it is what justifies it.