Diaconescu-Goodman-Myhill Theorem

Theorem
The axiom of choice implies the law of excluded middle.

Proof
Let $\mathbb B = \set {0, 1}$.

Let $p$ be a proposition.

Let the following two sets be defined:
 * $A = \set {x \in \mathbb B: x = 0 \lor p}$
 * $B = \set {x \in \mathbb B: x = 1 \lor p}$

where $\lor$ denotes the disjunction operator.

We have that:
 * $0 \in A$

and:
 * $1 \in B$

so both $A$ and $B$ are non-empty

Then the set:
 * $X = \set {A, B}$

is a set of non-empty sets:

By the axiom of choice, there exists a choice function:
 * $f: X \to \mathbb B$

since $\displaystyle \bigcup X = \mathbb B$.

There are four cases:


 * $(1): \quad \map f A = \map f B = 0$

This means that $0 \in B$.

But for that to happen, $\paren {0 = 1} \vee p$ must be true.

So by Disjunctive Syllogism, $p$ is true.


 * $(2): \quad \map f A = \map f B = 1$

This means that $1 \in A$.

Arguing similarly to case $(1)$, it follows that $p$ is true in this case also.


 * $(3): \quad \map f A = 1 \ne \map f B = 0$

This means that $A \ne B$ (or otherwise $f$ would pick the same element).

But if $p$ is true, that means:
 * $A = B = \mathbb B$

which is a contradiction.

Therefore in this case:
 * $\neg p$


 * $(4): \quad \map f A = 0 \ne \map f B = 1$

Using the same reasoning as in case $(3)$, it is seen that in this case:
 * $\neg p$

So by Proof by Cases:


 * $\paren {p \vee \neg p}$

That is the Law of Excluded Middle.

Also known as
The Diaconescu-Goodman-Myhill Theorem is also known as Diaconescu's Theorem‎ and the Goodman-Myhill Theorem.

It is ever 's endeavour to attest a theorem to as many contributors as appropriate.