Equivalence of Definitions of Minimally Inductive Class

Theorem
Let $A$ be a class.

Let $g$ be a mapping on $A$.

$(1)$ implies $(2)$
Let it be given that $A$ is inductive under $g$.

Let $A$ be a minimally inductive class under $g$ by definition 1.

Then by definition:
 * $A$ has no proper subclass $B$ such that $B$ is inductive under $g$.

Let $A$ have a subclass $C$ which is inductive under $g$.

Then by definition, $C$ is not a proper subclass of $A$.

Thus by definition of proper subclass:
 * $C = A$

By definition of subclass:
 * $A \subseteq C$

and:
 * $C \subseteq A$

That is:
 * $\forall x \in A: x \in C$

and:
 * $\forall x \in C: x \in A$

That is: $C$ contains all elements of $A$.

Thus $A$ is a minimally inductive class under $g$ by definition 2.

$(2)$ implies $(1)$
Let it be given that $A$ is inductive under $g$.

Let $A$ be a minimally inductive class under $g$ by definition 2.

Then by definition:
 * Every subclass of $A$ which is inductive under $g$ contains all the elements of $A$.

Let $C$ be a subclass of $A$ which is inductive under $g$.

By definition of subclass:
 * $C \subseteq A$

But by hypothesis:
 * $\forall x \in A: x \in C$

That is:
 * $A \subseteq C$

By definition of equality of classes it follows that:
 * $A = C$

and so by definition $C$ cannot be a proper subclass of $A$.

Thus $A$ is a minimally inductive class under $g$ by definition 1.

$(1)$ implies $(3)$
Let $A$ be a minimally inductive class under $g$ by definition 1.

Then by definition:

By definition of inductive class:


 * $\O \in A$
 * $A$ is closed under $g$.

Putting these definitions together, we have that:
 * No proper subclass of $A$ such that $\O \in A$ is closed under $g$.

That is, $A$ is minimally closed under $g$ with respect to $\O$

Thus $A$ is a minimally inductive class under $g$ by definition 3.

$(3)$ implies $(1)$
Let $A$ be a minimally inductive class under $g$ by definition 3.

That is, $A$ is minimally closed under $g$ with respect to $\O$.

That is:
 * $(1): \quad \O \in A$
 * $(2): \quad$ No proper subclass of $A$ such that $\O \in A$ is closed under $g$.

Hence by definition of inductive class:

Thus $A$ is a minimally inductive class under $g$ by definition 1.