I think it is true but I don't have a clue as to how I'm supposed to construct such a subset (working under ZFC)
2026-04-06 03:41:56.1775446916
Let $m$ be a cardinal number less than $q$. Prove that a set of cardinality $q$ contains a subset of cardinality $m$.
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Let $\kappa$ be the least ordinal of cardinality $m$ and $\lambda$ the least ordinal of cardinality $q$. Then $\lambda$ is a subset of $\kappa$. Now use the fact that any set of cardinality $m$ admits a bijection with $\kappa$, and any set of cardinality $q$ admits a bijection with $\lambda$.