Is the below language Non R.E?
Assume that $L$ is r.e. and $M_L$ recognizes $L$. Given $\langle M,w \rangle$, $M$ on $w$ either halts or not. This means that either $\langle M, w, 0 \rangle$ or $\langle M, w, 1 \rangle$ must be in $L$. Then using a universal Turing machine start to run/simulate $M_L$ on both inputs $\langle M, w, 0 \rangle$ and $\langle M, w, 1 \rangle$, simultaneously, say one step at a time for each input. After each step check if $M_L$ halts on at least one input. Since $L$ is r.e. and exactly one of $\langle M, w, 0 \rangle$ and $\langle M, w, 1 \rangle$ belongs to $L$, $M_L$ must eventually halt on one of them. Depending on which input $M_L$ halts, you can tell if $M$ halts on $w$. This decides the Halting problem and hence $L$ is not r.e.