Is this proof correct?

Question

Mathematical analysis

Prove A\(B\C)=(A\B)∪(A\C)
Proof:
x ∈ A ∖ (B \ C)
x ∈ A, provided that x ∉ B \ C
The last condition means that x ∉ B and x ∉ C (because it does not belong to both B and C).
Thus: (x ∈ A ∖ B) and (x ∈ A ∖ C)
That is, x ∈ (A ∖ B) ∪ (A ∖ C) Subtext
just proved:
x ∈ A ∖ (B \ C) ⟹ x ∈ (A ∖ B) ∪ (A ∖ C)

Can this proof be considered correct?

Answer 1

It is forbidden

x ∈ A, provided that x ∉ B \ C
The last condition means that x ∉ B and x ∉ C (because it does not belong to both B and C).

Take B=[2;5], C=[4;6], then B\C=[2;4), x=5, x is contained in both B and C, while x ∉ B \ C