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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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1 Answer

up vote 3 down vote accepted

$A\subseteq B$ 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 $A=B$.

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so, i can use biconditional to proof using logic? x<=>y is what i did, right? –  Matheus Silva Aug 11 '13 at 22:15
    
i think did not express myself very well, so, what i want, is to proof that A=B using A⊆B and B⊆A –  Matheus Silva Aug 11 '13 at 22:22
    
is that u post? if was, that SURE straightfoward...i had learn some different way...thats why i'm talking about roundabout... –  Matheus Silva Aug 11 '13 at 22:29
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