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I would really appreciate it if you could explain the set notation here

$$\{n ∈ {\bf N} \mid (n > 1) ∧ (∀x,y ∈ {\bf N})[(xy = n) ⇒ (x = 1 ∨ y = 1)]\}$$

1) What does $∀x$ mean?

2) I understand that $n ∈ {\bf N} \mid (n > 1) ∧ (∀x,y ∈ {\bf N})$ means $n$ is part of set $\bf N$ such that $(n > 1) ∧ (∀x,y ∈ {\bf N})$. What do the $[\;\;]$ and $⇒$ mean?

3) Prove that if $A ⊆ B$ and $B ⊆ C$, then $A ⊆ C$

I could prove it by drawing a Venn diagram but is there a better way?

5 Answers 5