``Hello!

I think you mean `A sub B` and `B sub C.` Yes, this implies that `A sub C.` Let's prove.

A set `A sub B` means by definition that any element `a in A` is also ` in B.` `B sub C,` in turn, means that any `b in B` is also `in C.`

Now whether `A sub C?` Any ` a in A` is also `in B` because `A sub B.` `a in B` implies `a in C` because `B sub C.` This way we proved that any `a in A` is also `in C,` so by definition `A sub C.`

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