I have the following definition:
Let $A,B$ be sets. We say that $A$ is a subset of $B$, denoted $A ⊆ B$, iff every element of A is also an element of B, i.e.
For any object $x$, x$ ∈ A ⇒ x ∈ B$.
We say that $A$ is a proper subset of $B$, denoted $A \subsetneq B$, if$ A ⊆ B$ and
$A\neq B$.
Note: I know that "$⊆$" is usually used to denote a proper subset and "$\subset$ " is used to denote a subset instead but I guess that is beyond the point of the question, I just copied the textbook content.
My question:
Logically speaking (according to the definitions) $A$ being a subset of $B$ shouldn't imply $A=B$ since that would mean for any object $x$, x$ ∈ A ⇒ x ∈ B$ but this isn't enough to consider $A=B$ as we would need as well for any object $y$, y$ ∈ B ⇒ x ∈ A$ however if $A$ is a subset of $B$ and $A \neq B$ then we would just denote is as a prober subset instead of a subset only, so I find this kind of contradicting.
Best Answer
By definition, $A$ is an proper subset of $B$ if $A \neq B$, so if $A$ is an improper subset of $B$, then $A=B$.
Note that $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.