FALSE: consistent first-order ZF has a unique model up to isomorphism

Under the stated consistency antecedent, F1 and F3 supply $M\models T_{\mathrm{Z}\mathrm{F}}$ with carrier injecting into $\omega$. F2 makes this carrier infinite. An infinite subset of $\omega$ can be enumerated in increasing order: after finitely many entries it has a least unused member, and each member is eventually reached since only finitely many natural numbers precede it. Composing this enumeration with the injection's inverse on its range gives a bijection $\omega \to M$. Thus $\mathrm{\mid }M\mathrm{\mid }={\mathrm{\aleph }}_0$.

Under A1 apply F4 with $\kappa ={\mathrm{\aleph }}_1$: $M$ is infinite, its size is ${\mathrm{\aleph }}_0\le {\mathrm{\aleph }}_1$, and the membership signature is finite. It gives $N⪰M$ of size ${\mathrm{\aleph }}_1$, which satisfies every sentence of $T_{\mathrm{Z}\mathrm{F}}$ by elementarity. An isomorphism $M\to N$ would be a bijection, contradicting ${\mathrm{\aleph }}_0\mathrm{\ne }{\mathrm{\aleph }}_1$. These are the promised two witnesses under the antecedent. Neither the use of F2 nor the extension argument identifies either internal membership relation with external membership or asserts well-foundedness.