User blog comment:LittlePeng9/Long hierarchies of functions (on my other blog)/@comment-11227630-20171031155258/@comment-80.98.179.160-20171113182505

Proof that $$|\omega_1+1|>|\omega_1|$$:

You have $$[0,1[$$ as $$A$$, and $$B$$ as $$A\cup\{2\}$$.

$$B$$'s cardinality is 1 more than $$A$$'s.