Is ( (A⊆B) & (B⊆A) ) → A = B a theorem?

Question

(It's probably important to state that I'm not a trained mathematician. Apologies if this question is too naive.)

The formula in question is $((A \subseteq B) \And (B \subseteq A))\rightarrow(A=B))$.

This formula is a plausible formalisation of a purported proof by Thomas Hobbes, in his /Leviathan/, of the "co-extensiveness" of natural law and civil law.

It seems intuitively valid and I haven't been able to find any counterexamples. But I'd like to know with certainty that it's true, if it's true. Two questions follow from this:

1) /is/ it a theorem? if it is, what is its proof, if this proof is straightforward?

2) what kind of course-material would I have to master in order to produce such a proof myself? Is that course-material available on the internet?

(With regard to question two: of course I'm only asking about the /most advanced necessary/ material; I assume none of you can know exactly how backward I am from the evidence of this short post.)

Answer 1

It is a theorem, and the proof is just very simple set theory:

The set relation $A\subset B$ is equivalent to the statement at the level of elements that $x\in A \Rightarrow x\in B$. Similarly, $A=B$ means that $x\in A\Leftrightarrow x\in B$.

To prove your theorem we need to show $x\in A\Leftrightarrow x\in B$, given the assumptions $A\subset B$ and $B\subset A$. Suppose that $x\in A$. Since $A\subset B$, we have $x\in B$. Thus we have proved $x\in A\Rightarrow x\in B$. From the other assumption we prove in the same way that $x\in B \Rightarrow x\in A$. Since we have proved the implications in both directions, we have proved that $x\in A \Leftrightarrow x\in B$, from our assumptions, i.e. that $A=B$.

Answer 2

Yes, it is a theorem. It's proof is very simple.

$A \subseteq B$ means every element of set A is in set B also. $A \land B$ (you wrote conjunction symbol as $\&$, but I prefer $\land$) means A is true and B is true. If I say $A \subseteq B \land B \subseteq A$, I'm saying every element of A is in B and every element of B is in A. Clearly, this means that sets $A$ and $B$ have exactly same elements.

You used $\rightarrow$, but I'd use $\leftrightarrow$. $A \rightarrow B$ means if A is true then B is also true, but $A \leftrightarrow B$ means if A is true than B is also true and if B is true then A is also true.

Therefore, $A \subseteq B \land B \subseteq A \leftrightarrow A = B$ means if every element of A is in B and every element of B is in A, then A and B are equal, and if A and B are equal, then every element of B is in A and every element of A is in B. If I used a right instead of a double arrow, then there would be a possibility that if $A = B$ is true $A \subseteq B \land B \subseteq A$ isn't true. In other words, $A \subseteq B \land B \subseteq A$ wouldn't need to be true for $A = B$ to be true.