For any set A, define the successor of A, denoted S (A), by S (A) = A ∪ {A}. One way to “construct” the natural numbers and zero from set theory is by the following correspondence: 0 ↔ ∅, 1 ↔ S (∅), 2...

For any set A, define the successor of A, denoted S (A), by S (A) = A ∪ {A}. One way to “construct” the natural numbers and zero from set theory is by the following correspondence:

0 ↔ ∅, 1 ↔ S (∅), 2 ↔ S (S (∅)), 3 ↔ S (S (S (∅))), …

Note that the axiom of infinity guarantees that all of these successor sets exist. Show that 5 ↔ {0, 1, 2, 3, 4}.