# Discrete Math proof equality

i need to prove this: if A⊆B and B⊆A so A = B

i know that, A⊆B => xEA->xEB.

B⊆A => xEBxEA.

How can i proof with A=B?

A = B only if A⊆B^B⊆A.

So, the result is:

(x∉A^X∈A)v(x∉A^x∉B)v(x∈B^x∉B)v(x∈B^x∈A)

False                False


so, it remains...

(x∉A^x∉B)v(x∈B^x∈A)


?? thats right?

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$A\subseteq B$ means that $x\in A\implies x\in B$. $B\subseteq A$ means that $x\in B\implies x\in A$. But $A=B$ means that $x\in A\iff x\in B$. Therefore, $A\subseteq B$ and $B\subseteq A$ if and only if $A=B$.