Proof of $\Bbb{N}$ is inductive

Question

Here there is a proof
but I think it is incomplete(missing $1\in\Bbb{N}$)

Note: A⊆R is inductive if and only if 1∈A and ∀x∈A⇒x+1∈A.
By definition, $\Bbb{N}$ is the intersection of all inductive sets

I tried to do a proof here:

$\Bbb{N} = \bigcap A_i\in I$ : $1 \in A_i \land \forall x\in A_i \Rightarrow x+1 \in A_i$

This is the same as:

$\Bbb{N} = \{x: x\in A_1 \land x\in A_2 \land x \in A_3 \land ...\}$

As $1 \in A_i\in I \Rightarrow 1\in \bigcap A_i\in I $ ie $1\in \Bbb{N}$

the rest is same

Is my proof correct?