About valuation and domination

Question

I'm trying to understand different formulation of domination. Taking a field $K$ and two surjectives valuations $v$ and $w$ on it, TFAE:

1. $v=w$
2. $\mathcal{O}_v=\mathcal{O}_w$
3. $\mathcal{O}_v\supseteq\mathcal{O}_w$ and $\mathfrak{m}_v\cap\mathcal{O}_w=\mathfrak{m}_w$
4. $\mathcal{O}_v\supseteq\mathcal{O}_w$ and $\mathfrak{m}_v\supseteq\mathfrak{m}_w$

I think that's ok with that.

I'd like to have some description with uniformizer, for example

5. For all uniformizer $t_w$ for $w$ one has $v(t_w)>0$
6. For one uniformizer $t_w$ for $w$ one has $v(t_w)>0$

But such things would be equivalent with

7. $\mathfrak{m}_v\supseteq\mathfrak{m}_w$

Is that true, how prove that?