Is subset property transitive?

Question

Is subset property transitive? by that I mean if $X$ and $A$ are sets and $A\in X$ then is every subset of $A$ also a subset of $X$?

Simple example $A=\{0,1\}$ , $X=\{A\}$ then is it true that $0,1 \in X$?

Answer 1

Notice that $\subseteq$ and $\in$ means different thing. $\subseteq$ means subset while $\in$ means element of.

Subset is a transitive property in the sense that if $A \subseteq B$ and $B \subseteq C$ , then $A \subseteq C$.

If $A \in X$, then every subset of $A$ need not be a subset of $X$. For example Let $X = \{A\}$ and $A=\{1,2\}$, then $A \in X$, but $\{1\}$ is not a subset of $X$.

For your example, $X=\{A\}$, then $A \in X$, however, $0 \notin X$, also $\{0\} \notin X$. $X$ only has one element which is $A$ and $X$ has two subsets which are $\emptyset$ and $X$.