Is the theory $TA+\lnot Con(TA)$ consistient?

In particular, for every $TA \vdash \phi$, we take as an axiom $\phi$, and $TA \vdash \phi$. We also assert $TA \vdash 0 = 1$. We call this theory $TA + \lnot Con(TA)$.

Note that the theory we are talking about does not include the statement that $TA$ is true (i.e. $TA \vdash \phi \iff \phi$). The theory can only see what $TA$ implies, and that $TA$ is a theory of first order logic (specifically, a weak set theory).

Is this theory consistent?