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New to math SE. I'm working through a book for fun on set theory but am still trying to wrap my head around methods of proving things. They've given a couple of very simple statements to prove, but beyond the intuitive notion of "if A is a subset of B and B is a subset of C then A is a subset of C" (I'm picturing marbles in my head, which is attaching incorrect context to the notion) I can't see strictly speaking what the proof is.

The only axiom given so far is the Axiom of Extentionality which I don't think applies because it implies A ⊆ B and B ⊆ A (which isn't stated. It has also given me the subset definiton A ⊆ B iff x ∈ A ⇒ x ∈ B. But again, beyond intuition, I don't know how to get there. Can anyone point me in the right direction?

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    Use the def : if $x \in A$ then ... using $A \subseteq B$ and $B \subseteq C$. – 2017-01-26
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    you have to say x belongs to A implies x belongs to C. Suppose x belong to A, since A subset B by definition we can say x belongs to B, again B subset C, again by definition x belongs to C – 2017-01-26
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    Ack, you're right. Thanks for humoring me. I'm new to this and it's difficult to switch how my brain normally works when doing proofs. – 2017-01-26
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    yeah thinking in the formal way is the difficult thing for a beginner in any science subject, as we were not used to it much @Xedni – 2017-01-26

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