Cross price elasticity of demand vs Price elasticity of demand

Is the cross price elasticity of demand between two goods always lower in magnitude than the price elasticities of demand of each good?

For example if the utility function is $$U=\min\{x,y\}$$, then the demand function for x is given by $$D=\frac{M}{P_x+P_y}$$.
The own price elasticity of demand for x is: $$-\frac{M}{(P_x+P_y)^2} \frac{P_x}{x}$$.
The cross price elasticity of demand for x is: $$-\frac{M}{(P_x+P_y)^2} \frac{P_y}{x}$$.
If these elasticities are evaluated at $$P_x=P_y=p$$, then for any value of x, you have the same elasticities.