- #1
evinda
Gold Member
MHB
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Hello! (Cool)
Sentence
The set, that does not contain any element, is unique.
Proof:
Let's suppose that $a,b$ are sets, so that each of these sets does not contain any element and $a \neq b$.
From the axiom: Two sets, that have the same elements, are equal., there is (without loss of generality )
$$x \in a \text{ and } x \notin b (*)$$
$a,b$ do not contain any element.
$$\forall x (x \notin a)$$
$$\forall x (x \notin b)$$
$$\forall x (x \notin a \leftrightarrow x \notin b) (**)$$
From $(*)$ and $(**)$, we have a contradiction, so the set that does not contain any element is unique.Could you explain me how we get the relation $(**)$ ?
Sentence
The set, that does not contain any element, is unique.
Proof:
Let's suppose that $a,b$ are sets, so that each of these sets does not contain any element and $a \neq b$.
From the axiom: Two sets, that have the same elements, are equal., there is (without loss of generality )
$$x \in a \text{ and } x \notin b (*)$$
$a,b$ do not contain any element.
$$\forall x (x \notin a)$$
$$\forall x (x \notin b)$$
$$\forall x (x \notin a \leftrightarrow x \notin b) (**)$$
From $(*)$ and $(**)$, we have a contradiction, so the set that does not contain any element is unique.Could you explain me how we get the relation $(**)$ ?