Partial order of a finite set is an intersection of a finite number of linear orders of this set

Question

I am trying to prove that every partial order of a finite set is an intersection of a finite number of linear orders of this set. Can this be proved using these observations:

1. a partial order has a linear extension
2. this partial order is an intersection of its linear extensions
3. there is a finite number of linear extensions

Or is there a simpler proof?