Questions about logic and proof systems

Question

I encountered two similar expressions when prooving two sets A and B are equivalent.

For $A \subseteq B$, we have two ways to prove?

$\forall x \in A$, then $x \in B$

$\forall(x \in A \longrightarrow x \in B)$

1. Are these two ways both right?

2.how to deal with the situation when A is $\varnothing$ coz in the first way we suppose $A$ is not empty?

Appreciate for your helping hands.

Answer 1

Both conditions mean that $\forall x(x\in A\Rightarrow x\in B)$. Forall and implication belong together im math. formulations.

Answer 2

Actually there is no problem with the empty set; this is because $p\implies q$ is true whenever $p$ is false, so if $A=\varnothing$ then $x\in A$ is false and because of what I said above we have that the empty set is a subset of every set!