Full Idea
Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets.
Gist of Idea
Sets always exceed terms, so all the sets must exceed all the sets
Source
Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127)
Book Reference
Russell,Bertrand: 'Essays in Analysis', ed/tr. Lackey,Douglas [George Braziller 1973], p.127
A Reaction
The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom.