Herewith, the Schroeder-Bernstein theorem without the pigeonhole principle and without explicit cardinal numbers.
We use the dot notation for "and" as well as the logical product of u and v expressed uv or of (u) and (v) expressed (u)(v). The symbol ≺ means "less numerous than," which certainly implies the concept of cardinality. The symbol ≺≈ means "less numerous than or equal to." The symbol ~ means "not."
The notation of statement 0 seems to imply the conclusion. Yet, sets can be peculiar things. Formal proof that the conditions for x and y imply that they are bijective is required.
0.
x ≺≈ y • y ≺≈ x • --> • x ≈ y
To prove.
1.
x ≺ y • x ≈ y
Assumption
2.
x ≈ r ⊂ y • ~(r ≈ y)
Same as 1.
(The proper subset r is chosen such that it is not
bijective with y,
which could happen for r infinite.)
3.
r ⊂ y • ~(r ≈ y) : x ≈ y
Assumption
3a.
y ≈ s ⊂ x • ~(s ≈ x)
Assumption
4.
x ≈ r • ~(r ≈ y)
2, 3
5.
~(x ≈ y)
MP on 4
6.
~(x ≺ y • x ≈ y)
Deduction Theorem
on 1 to 5.
7.
~(y ≺ x • x ≈ y)
Steps 1-6, substitution of y for x
8.
x ≺ y • y ≺ x : ~(x ≈ y )
Assumption
9.
x' ≈ x • (x', r, y disjoint)
Condition
(For example let x' = x X 0
if no members of r and y are so constructed.
It is always possible to establish 9.)
10.
y' ≈ y • (y', s, x disjoint)
Condition
11.
x' ∪ r ≈ y
2, 9
12.
y' ∪ s ≈ x
3, 10
13.
x' ∪ r • ≈ • y' ∪ s
Substitution
14.
y ≈ x
Substitution
15.
~[(x ≺≈ y)(y ≺≈ x) ~(y ≈ x)]
Deduction theorem 8-15
(Assumption 8 is inconsistent and hence
false.)
16.
x ≺≈ y • y ≺≈x --> x ≈ y
Deduction theorem 1-15
(Statement 0 is the only scenario left.
One might ask what about a case in which none of
the scenarios holds. But then one would to argue with one or
more of the axioms of the standard set theories.)
In an earlier draft, the step numbered 3a was omitted. Rather than renumber the proof, I chose to use that alphanumeric form. And, as I did not indicate which steps were subsidiary to others, 3a should not be read to indicate a special logical relationship.
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