It may be necessary to look at both the formal definition (right below) and the text explanation on web pages (informal) to catch the idea. Axiom of Foundation is one fast rabbit.
∀S (S = ∅ → (∃x ∈ S) S ∩ x = ∅)
We might have it below:
- everything is a set
- a nonempty set must be a “wrapper” around a set, so the generic {A} qualifies for the general case
- {A} contains A.
- {A} ∩ A = ∅
We think we can say that the last line is the whole point of the axiom. For two sets to be equal, they must share everything in common; to have a null intersection is to have nothing in common, therefore, {A} and A are not equal.
Coming soon: they said that Foundation and Pairing give us that “no set is an element of itself.”
Rob Sterling 