How Can We Prove Basic Properties of Set Theory?

[evinda]

Hello! (Wave)

I want to prove the following sentences:

1. $A \subset B \wedge B \subset C \rightarrow A \subset C$
2. $A \subset B \wedge B \subset A \rightarrow A=B$
3. $\varnothing \subset A$
4. $\varnothing=\{ x: x \neq x \}$

That's what I have tried:

1. $A \subset B \leftrightarrow \forall x(x \in A \rightarrow x \in B)$
   $B \subset C \leftrightarrow \forall x(x \in B \rightarrow x \in C)$
   
   $A \subset B \wedge B \subset C \leftrightarrow \forall x(x \in A \rightarrow x \in B \rightarrow x \in C) \leftrightarrow \forall x(x \in A \rightarrow x \in C) \leftrightarrow A \subset C$
   $$$$
2. $A \subset B \leftrightarrow \forall x(x \in A \rightarrow x \in B)$
   $B \subset A \leftrightarrow \forall x(x \in B \rightarrow x \in A)$
   $A \subset B \wedge B \subset A \leftrightarrow \forall x(x \in A \leftrightarrow x \in B)$
   $$$$
3. In my notes, there is the following proof for this sentence:
   
   Let $x$ an element that does not belong to $A$. Then, since $\varnothing$ does not contain any element, $x$ does not belong to $\varnothing$.
   
   Why do we conclude that $C \subset D$, showing that if $x \notin D \rightarrow x \notin C$ ?
   $$$$
4. How can we prove that $\varnothing=\{ x: x \neq x \}$ ?

 

[Valkarie]

To prove that $\varnothing=\{ x: x \neq x \}$, we can use the proof by contradiction. Assume that $\varnothing \neq \{ x: x \neq x \}$. Then, there must be an element $x$ such that $x \in \varnothing$ and $x \in \{ x: x \neq x \}$. This is a contradiction since $x \neq x$. Thus, we can conclude that $\varnothing = \{ x: x \neq x \}$.