We say $\succsim$ represents weak monotonic preferences if $$x,y \in X, \,\, y >> x \implies y \succ x $$
where $y >> x$ means that every element of $y$ is greater than every element of $x$.
And we say $\succsim$ represents strong monotonic preferences if $$x,y \in X, \,\, y \geq x, y \neq x \implies y \succ x$$
where $y \geq x$ means that at least one element of $y$ is greater than an element of $x$ and all others are equal.
My question is: do strong monotonic preferences imply weak monotonic preferences?
My answer: yes. The reasoning is as follows:
Since the set $A = \{ x,y \in X: y >> x \}$ is a subset of $B = \{x,y \in X: y \geq x, y \neq x \}$, then if $B \implies y \succ x$ it must also be true that $A \implies y \succ x$ and thus strong monotonic preferences imply weak monotonic preferences.
In plain English, if a bundle with more of one commodity and the same of all others is preferred, then a bundle with more of every commodity must also be preferred.
Is my reasoning correct?
Thanks!