NF+swf-Separation is inconsistent.
Let P1(X) be the set of one element subsets of X. Let z={Ø,{Ø},{{Ø}}} Let S={{P1(A),z}| A is infinite}.
Let T be the set of subsets of S.
(1) Suppose A is infinite. Then {P1(A),z}∩P1(A)=Ø and therefore swf({P1(A),z}).
(2) Suppose A is infinite. Then {P1(A),z}∩S=Ø. and therefore swf(S).
Proof: P1(A) is not in S because every element of S is a set with 2 elements.
z is not in S because every element of S is a set with 2 elements.
(3) S∩T=Ø.
Proof: Suppose A is infinite. Then {P1(A),z} is not a subset of S.
Let P={{a,b}| a∈T and b∈S}.
(4) Suppose a∈T, b∈S, and b∉a. Then a∩{a,b}=Ø and b∩{a,b}=Ø, and thus swf({a,b}).
Proof: If a∈a, then a∈S, and consequently there is an infinite set A with
P1(A)∈a. But every element of a is a 2 element set. Therefore a∩{a,b}=Ø.
a∉b because neither element of b is a set of 2 element sets. b∉b because
b is a 2 element set with no element which is a 2 element set.
Therefore b∩{a,b}=Ø.
(5) Suppose a∈T and b∈S. Then {a,b}∩P=Ø and therefore swf(P).
Proof: By (3), a∩T=Ø and so a∉P. P1(A)∈b for some infinite set A. By (2), P1(A)∉S.
P1(A)∉T because T is a set of 2 element sets. Therefore b∉P.
(5) There is a 1-1 function from T to S.
Proof: Let N be the set of natural numbers and let O be the set of odd numbers.
Let d be the 1-1 function from sets to infinite sets defined by
da={2n|n∈N∩a}U{x|x∈(a-N)}UO.
Now define f:T-->S by ft={P1(d{A|{P1(A),z}∈t}),z}. Then f is 1-1.
Let f be a 1-1 function from T to S. Let F={{a,b}|a∈T and b∈S and fa=b}. By (4), swf(F).
Let φ(x) be the formula ∃t(t∈T∧∃p(p∈F∧t∈p∧x∈p∧x∉t)). Then for x∈S, φ(x) is equivalent to its
relativization to swf. By swf-Separation, there is a C such that x∈C<-->x∈S∧φ(x).
Suppose fC=c. Then c∈C<-->c∉C.