User:Dfeuer/CBS experiment

Theorem
Let $A$ be a class.

Let $B \subseteq A$.

Let $f: A \to B$ be an injection.

Then there is a bijection $g: A \to B$.

Proof
Let $Q$ be the class of all subsets of $A$ (sets which are subclasses of $A$).

For any class $X$, let $E(X)$ be an abbreviation for $A \setminus (B \setminus f[X])$.

Note that if $X$ and $Y$ are classes and $X \subseteq Y$ then $E(X) \subseteq $E(Y)$.

Let $P = \{p: p \subseteq A \land p \subseteq E(p) \}$.

Let $W = \bigcup P$.

Let $x \in W$.

Then there is a $p \in P$ such that $x \in p$.

Thus $x \in E(p)$.

Then $x \in E(W)$.

So we see that $W \subseteq E(W)$.

Thus $E(W) \subseteq E(E(W))$.

We wish to show that $E(W) \subseteq W$; but it's far from obvious whether this is actually true. Might be better to stick to the ugly bouncing approach.