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Feb 23 at 17:36 vote accept Dominic van der Zypen
Feb 23 at 10:31 history edited bof CC BY-SA 4.0
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Feb 23 at 9:31 comment added Emil Jeřábek The statement with a well-orderable family $\mathfrak S$ is equivalent to the axiom of choice for families of finite sets, just like (S$_3$). This is shown in my answer to the MO question linked from the OP.
Feb 23 at 8:55 history answered bof CC BY-SA 4.0