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Order 13 of the following sentences so that they form a logical proof of the statement:

For all non-empty sets $A$ and $B$ , $A \times B = B \times A \Rightarrow A = B$ .

Choose from these sentences:

$\forall b \in B$ , $(x,b) \in A \times B$
We have shown $y \in B \implies y \in A$
Let $x \in A$
Assuming $A = B$ , let $a \in A$ and $b \in B$
I am so tired of this proof!
For any $a \in A, (a,y) \in A \times B$
$A \times B = B \times A \implies (x,b) \in B \times A$ and so $x \in B$
Hence $A = B$
Next, prove $B \subset A$ by assuming $y \in B$
Therefore $B \subset A$
Therefore $A \subset B$
Let $A \times B = B \times A \Rightarrow A = B$
It's obvious.
We have shown $x \in A \implies x \in B$
Assume $A \times B = B \times A$ for nonempty sets $A$ and $B$ .
$A \times B = B \times A \implies (a,y) \in B \times A$ and so $y \in A$
To show $A = B$ , first prove $A \subset B$ and then prove $B \subset A$
$A = B \Rightarrow A \times B = B \times A$

Your Proof:

Choose from these sentences:

$\forall b \in B$ , $(x,b) \in A \times B$
We have shown $y \in B \implies y \in A$
Let $x \in A$
Assuming $A = B$ , let $a \in A$ and $b \in B$
I am so tired of this proof!
For any $a \in A, (a,y) \in A \times B$
$A \times B = B \times A \implies (x,b) \in B \times A$ and so $x \in B$
Hence $A = B$
Next, prove $B \subset A$ by assuming $y \in B$
Therefore $B \subset A$
Therefore $A \subset B$
Let $A \times B = B \times A \Rightarrow A = B$
It's obvious.
We have shown $x \in A \implies x \in B$
Assume $A \times B = B \times A$ for nonempty sets $A$ and $B$ .
$A \times B = B \times A \implies (a,y) \in B \times A$ and so $y \in A$
To show $A = B$ , first prove $A \subset B$ and then prove $B \subset A$
$A = B \Rightarrow A \times B = B \times A$

Your Proof: