Splitting turns an uncountable branch into an antichain

For each $\alpha <{\omega }_1$, cofinality gives $v\in b$ of height at least $\alpha$. If its height is greater, let $u$ be its unique predecessor of height $\alpha$; otherwise set $u=v$. Every $w\in b$ is comparable with $u$: if $w{\le }_{T}v$, use common-predecessor comparability, and if $v{<}_{T}w$, use $u{\le }_{T}v{<}_{T}w$. Maximality of $b$ therefore puts $u$ in $b$. Distinct nodes of the same height cannot both belong to a chain. Thus there is a unique $b_{\alpha }\in b\cap T_{\alpha }$ for every $\alpha$. For $\alpha <\beta$, comparability and height give $b_{\alpha }{<}_T{b}_{\beta }$.

The node $b_{\alpha +1}$ is an immediate successor of $b_{\alpha }$, since an intermediate node would have height strictly between $\alpha$ and $\alpha +1$. Conversely every immediate successor $u$ of $b_{\alpha }$ has height $\alpha +1$: if its height were larger, its predecessor at $\alpha +1$ would lie strictly between $b_{\alpha }$ and $u$. Since $\alpha +1<{\omega }_1$, splitting makes $S_{\alpha }=\{u:u\text{ is an immediate successor of }{b}_{\alpha },u\mathrm{\ne }{b}_{\alpha +1}\}$ nonempty. AC gives $a_{\alpha }\in S_{\alpha }$ for every $\alpha <{\omega }_1$.

If $\alpha <\beta$, then $b_{\alpha +1}{\le }_T{b}_{\beta }{<}_T{a}_{\beta }$. Were $a_{\alpha }$ and $a_{\beta }$ comparable, their different heights $\alpha +1<\beta +1$ would force $a_{\alpha }{<}_T{a}_{\beta }$. The two distinct nodes $a_{\alpha },b_{\alpha +1}$ of height $\alpha +1$ would then be predecessors of $a_{\beta }$, contradicting uniqueness at that height. Thus $a_{\alpha },a_{\beta }$ are incomparable.

Consequently $\{a_{\alpha }:\alpha <{\omega }_1\}$ is an antichain. Its indexing is injective because $\mathrm{ht}(a_{\alpha })=\alpha +1$, so it has cardinality ${\mathrm{\aleph }}_1$. The argument includes $\alpha =0$ and adjacent levels; all successor levels used remain below ${\omega }_1$.