= Dominance from a nonzero column antisymmetrizer
Let $t$ be a <Young tableau> of shape $\lambda$, and let $u$ be a <Young tableau> of shape $\mu$, both on $n$ entries. If the <Column antisymmetrizer of a Young tableau> $b_t$ satisfies $b_t\{u\}\ne0$ over any <field>, then $\lambda$ dominates $\mu$ in the <dominance order on partitions>.
If two entries from one column of $t$ lie in the same row of $u$, their transposition fixes the <tabloid> $\{u\}$. Pairing terms of $b_t$ by this transposition gives $b_t\{u\}=0$, including in <characteristic> two. Otherwise the first $r$ rows of $u$ contain at most $\min(r,\lambda'_j)$ entries from column $j$ of $t$. Thus
$$
\sum_{i=1}^r\mu_i\leq\sum_j\min(r,\lambda'_j)=\sum_{i=1}^r\lambda_i
$$
for every $r$, proving dominance.
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