Andrei1
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Prove that for all $$x,y\in\omega,\ \ x\subset y\vee y\subset x.$$
If I assume that the conclusion is false then I can prove that for some $$a\in x,\ b\in y$$ we have $$a\notin b$$ and $$b\notin a.$$
Also I am thinking that if assume the contrary then $$\omega$$ minus $$\{x\}$$ or minus $$\{y\}$$ or both is a smaller successor set. Should I try to prove this?
I get stuck in trying to prove for sets from $$\omega$$ the equivalence: $$a\subseteq b\wedge a\not=b\Leftrightarrow\exists c(a\cup c^+=b)$$.
If I assume that the conclusion is false then I can prove that for some $$a\in x,\ b\in y$$ we have $$a\notin b$$ and $$b\notin a.$$
Also I am thinking that if assume the contrary then $$\omega$$ minus $$\{x\}$$ or minus $$\{y\}$$ or both is a smaller successor set. Should I try to prove this?
I get stuck in trying to prove for sets from $$\omega$$ the equivalence: $$a\subseteq b\wedge a\not=b\Leftrightarrow\exists c(a\cup c^+=b)$$.