Why is the Russell Paradox necessary?

In ordinary set theory, in which Russell's Paradox is used
to show that assuming the existence of a set of all sets
leads to that paradox, ie. a contradiction, I wonder why
the Russell Paradox method is necessary.

We know that the cardinality of the set of all subsets of a
given set is greater than the cardinality of the given set.
But if we assume the existence of a set of all sets, by
definition it contains all of its subsets. Therefore the
cardinality of the set of all subsets of the set of all sets
cannot be greater than the cardinality of the set of all
sets, which is a violation of what we know about
cardinality. Hence there is no set of all sets.

.

Relevant Pages

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