[Math] Is {∅} a subset of the power set of X, for every set X

Question

I am reading the book "A Transition to Advanced Mathematics" 2014 edition, written by Smith, Eggen and St. Andre.

For every set X:

∅ ⊆ X

∅ ∈ power set of X

∅ ⊆ power set of X

{∅} ⊆ power set of X

Is {∅} ⊆ power set of X wrong?

I am an undergraduate student. Please help me!

The definition of power set of X is the set whose elements are subsets of X. (page 90 of the book)

I have this doubt because I remember that I read somewhere that ∅ ≠ {∅} because a set with the empty set is not empty.

Thanks for your answers!

What about this question:

If the definition of power set of X is the set whose elements are subsets of X, and {∅} is not a subset of X. How {∅} can be a subset of the power set of X? Is not that an contradiction?

Answer 1

Recall that $A\subseteq B$ if whenever $a\in A$, then we have that $a\in B$ as well.

Since $\varnothing\in\mathcal P(X)$, it means that every element of $\{\varnothing\}$ is also an element of $\mathcal P(X)$, so $\{\varnothing\}\subseteq\mathcal P(X)$ is indeed correct.