Proving $A \in B \land B \subseteq C \to A \in C$.

Question

We are given $A \in B \land B \subseteq C \to A \in C$, how to proove that it is true for any $A,B,C$?

I am having trouble proving it correctly... This is what i am thinking:

• $A \in B$ means $B\{ A$, other elements from $B\}$
• $B \subseteq C$ means $C\{ A$, other elements from $B$, other elements from $C\}$
• $A$ belongs to $C$, so it is true.

Is this correct? and how to write every elements excluding $x$?

Answer 1

Your attempt at proving it is not clear to me, so I will not comment much on its accuracy. It seems like you have a rough understanding of the underlying notions though.

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Suppose $A\in B$ and $B\subseteq C$. By definition of $B\subseteq C$, for all $b\in B$, we have $b\in C$. Let $b=A$. Thus $A\in C$.

Answer 2

In set-theory $\in$ is a primitive notion that has no definition.

It is just some kind of relation that two sets have or have not: $$B\in C\text{ or }B\notin C$$

Based on $\in$ a new relation $\subseteq$ is defined by stating that: $$B\subseteq C\iff x\in C\text{ for every }x\text{ that satisfies: }x\in B $$

So - by definition - if $B\subseteq C$ then $A\in C$ is a true statement whenever $A\in B$.

This comes to the same as saying that: $$A\in B\wedge B\subseteq C\implies A\in C$$is a true statement.

So here a proof is actually nothing more than the application of a definition.