the class of all models

Explain me please, why if the set of first-order sentences has an infinite model, then the set (class) of all models of , , is strictly bigger than any set ?

Here is a passage from a textbook:

Quote:

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... is sometimes unbounded. It is unbounded precisely when has an infinite model. By unbounded we mean that for any set , is strictly bigger than ...

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This probably has to do with the (upward) Löwenheim–Skolem Theorem, according to which Γ has models of every infinite cardinality. Then the question reduces to showing that the class containing all cardinals is bigger than any set.