Let $\hat{V}$ be the standard universe constructed as
$ \hat{V} = V_0\cup V_1 \cup V_2 \cup ....$
where
$V_0$ is the set of primary elements and
$V_{v+1} := V_v \cup P(V_v)$
For the following (failed) proof we assume $V_0 = \mathbb{N}$.
Given then whole non-standard course we also got a transfer-function and a transfer-principle.
Now, I was trying to transfer the statement "Every subset of $\mathbb{N}$ has a smallest element", as formula:
$(\forall A\in V_2)\bigg(A\subset\mathbb{N} \Leftrightarrow (\exists a\in A)\big((\forall b\in A)(a\le b)\big)\bigg)$
The magic $V_2$ is used there, as $V_2 = P(P(V_0))$ contains all subsets of $\mathbb{N}$ as elements.
Given the transfer principle, this should transfer to a formula where all standard-elements are exchanged with non-standard-elements, in this case
$V_2 \,\text{ by } \,^*V_2\quad$and $\quad\mathbb{N}\,$ by $\,^*\mathbb{N}\quad$ and$\quad$ $\le$ $\,$ by $\,$ *≤
Finally, the transferred formula should look like this:
$(\forall A\in {^*V_2})\bigg(A\subset\mathbb{^*N} \Leftrightarrow (\exists a\in A)\big((\forall b\in A)(a $ *≤ $ b)\big)\bigg)$
It looks right, but as $\mathbb{^*N}$ contains infinitely infinite elements, I can construct a subset $\mathbb{^*N}-\mathbb{N} \subset \mathbb{^*N}$ that "defies" the proof.
E.g. say $x\in\mathbb{^*N-N}$ is the smallest element of the set. Then $x-1$ is smaller, yet infinite, and as such member of $\mathbb{^*N-N}$.
So, somewhere along the line I must have made a mistake or (more probable) misunderstood something. But what is it?