begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §4. FINITE AND INFINITE SETS. "end quote
These pages assume that the reader has the book and is reading along.
topic index     contents
Please read these notes from the beginning to here and then the first 3 sentences of paragraph 4 of §4.

PROPER SUBSETS OF INFINITE SETS

First please recall that the meaning of the material implication     is   ,   where   ~   means   "not"   and where the symbol     means "and".   Now recall that all important substitution exercise that was first encountered on the subset page.   Replace   A   above by   ~B   and replace   B   by   ~A.   We get the following:

~B~A       means       ~(    ~B ~(~A)   ).

But   ~(~A)   is just   A   and the symbol     is commutative, which means we can read the "and" statement either from right to left or from left to right. Consequently we see that   "not B implies not A"   has exactly the same meaning as   "A implies B".   Rosenlicht uses this here:

begin quote ita-1-4-4-pg11 "
It is easy to show that a set   X   is infinite if and only if it may be put into one-one correspondence with a proper subset of itself. To do this, note first that if   X   is finite then any proper subset has a smaller number of elements, whereas two finite sets in one-one correspondence must have the same number of elements. This proves the   "if"   part.
"end quote

The   if part   is the statement   "If   X   can be put into one-one correspondence with a proper subset of itself, then   X   is infinite." To use the result presented above, we negate and switch:   "If   X   is finite, then it can't be put into one-one correspondence with a proper subset of itself."   This is what Rosenlicht proves.