About distributivity of a semiring

Question

I know that if $(R, +, ., 0, 1)$ is an idempptent semiring and $x, y\in R,$ then by distributivity of $\cdot$ over $+$, we can write $x+xy+x=x(1+y+1)=x.1=x.$ This time, if my semiring is $(P(X), \cup, \cap, \lbrace\emptyset \rbrace, X)$. Then for all $X_1, X_2\in P(X)$, and replacing $+$ by $\cup$, $\cdot $ by $\cap$ and $1$ by $X$ in the above equation, we get $X_1\cup (X_1\cap X_2)\cup X_1=X_1\cap \lbrace X \cup X_2\cup X \rbrace=X_1\cap X=X_1,$ where $\cap$ distributes over $\cup$, and $P(X)$ is a power set of $X$. Is this equation correct?