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.

Is this theory consistent?

Note: TA + $\lnot Con(TA)$ is not a theory in the language of arithmetic. We are adjoining a symbol for $TA \vdash$ to the language. You can also interpret it as a weak set theory.