I am asked to prove the following: Working in ZF - P.
If $M$ is (Edit) a transitive model of ZF - P, and $R, A \in M$, then
$(R \ well \ orders \ A) ^M \rightarrow (R \ well \ orders \ A)$
I am struggling with the minimal element. Since $R$ well orders $$A relative to $M$ we have that: $\forall X \in M s.t. X \subset A (X \neq 0 \rightarrow \exists y \in X (\lnot \exists z \in X (zRy))]$
So I need to show that for any $X$ (not necessarily in $M$) s.t. $X \subset A$ and $X \neq 0$, X has a minimal element.
My idea was to suppose there is such an X with no minimal element and arrive at a contradiction by showing this would imply A has no minimal element but the details of this are tripping me up.
Considering any element $a \in A$ we try to show $\exists b \in A$ s.t. $b R a$.
If $a \in A$ then $a \in X$ or $a \in A-X$. If $a \in X$ then we are done since X has no minimal element so we can always find something smaller.
Now if $ a \in A-X$ then...I want to say something in $X$ is smaller then it? Or possibly compare $a$ and elements in $X$ since those are both in $M$ and hence well ordered there, but I'm getting stuck.
Any assistance would be awesome.