For each of the following sets A, B prove or disprove whether A ⊆ B and B ⊆ A

Would this proof be sufficient for just proving B ⊆ A alone?

$A = {x ∈ Bbb Z : ∃y∈Bbb Z,x = 5y + 1}, $$B = {x ∈ Bbb Z : ∃y∈Bbb Z, x = 10y − 9}$

Suppose some arbitrary element x is in B. if x∈B, then x∈A by definition of B ⊆ A.
Also, by the definition of set B,

x = 10y – 9 for some y
= 10y – 9 + 10 -10
= 10y – 10 + 1
= 5(2y-5) + 1

Since, 2y-5 is an integer, x is in A by definition.
Therefore B ⊆ A.