how-to-prove
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Not correct definition is used
https://github.com/psibi/how-to-prove/blob/c17d7a74473e73e3428c666441cd98be7ea67579/chapter%204/section4-4.tex#L602
In this line you say that UF is in F. But this is not true since if we have F = {{1,2,3,4}, {5}}, UF = {1,2,3,4,5} and {1, 2, 3, 4, 5} is not in F.
Free feel to send PR fixing this! Thanks!