Definition: $A\cap B=\{x | x\in A \quad \text{and} \quad x\in B\}$
My problem appears when I try to apply this definition to empty sets as following:
$$\emptyset\cap \emptyset=\{x | x\in \emptyset \quad \text{and} \quad x\in \emptyset\}$$
Then I evaluate the proposition "$x\in \emptyset \quad \text{and} \quad x\in \emptyset$". If $x\in\emptyset$ is false then this has $0$ value.
That is:
$$x\in \emptyset \quad \text{and} \quad x\in \emptyset\equiv 0 \wedge 0\equiv 0$$
So what does $\emptyset\cap \emptyset=\{x \;| \; 0 \}$ mean?
In addition, generally, how we should consider the proposition written in the feature of the set $\{x\;|\; PROPOSITION \}$ when it comes proposition takes the value $0$ or $1$?