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Proof. ##~~\rightarrow ~~ ## Suppose ##N## is normal such that ##N \not\in N## but since ##N## contains all normal sets, it should include itself since it is a normal set by assumption. Thus ##N \in N##

##\leftarrow~~## The argument is the same as before. QED

Can anyone help me check my proof? Also, is it correct to think that the crucial ingredient/key phrase here is "N contains all the normal sets"?