By " subset of S " I mean here : a set that is included in set S. ( I do not restrict " subset " to " proper subset").
In order to show that the asssertion is false, I can think of the following :
Suppose S is a set with no subset
(1) either this is set infinite , or finite
(2) if S is infinite, there is at least one proper subset of S such that this subset is equinumerous to S; hence, S has at least one proper subset, and, a fortiori, one subset
(3) If S is finite with cardinal n ,
then the cardinal of P(S) is 2 to the nth power.
in case S has no subset , the cardinal of P(S) is 0
so n should satisfy the equation : 2 to the nth power equals 0. ( That is : 2^n=0)
but there is no cardinal number satisfying this equation; hence an impossibility.
So, the set with no subset is neither infinite nor finite.
Consequently, there is no set with no subset.
The empty set is a subset of every set.
The empty set is a proper subset of every set except the empty set.