In this paper we prove:
Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.
Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.
Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.
The proof of Theorem 2 will follow from the lemmas which we establish below.
We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.
Clearly, in every theory of sets we have:
(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),
(2) (x = y) ↔ (P(x) = P(y)),
(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),
(4) ⋃P(x) = x.
In view of (2), (3) and the definition of equality, we have:
Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.
We have also, in view of (4):
Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.