Let as say that a lattice $L$ has the anti-blocking property if for all $a,b,x\in L$ such that $a \nleq b$, $a \nleq x$ and $x \nleq b$, either $a \nleq x \vee b$ or $a \wedge x \nleq b$ holds.
The intuition behind it, is that if I have $a \nleq b$, and I have some $x$ not greater than $a$
and not smaller than $b$, I can either increase $b$, or decrease $a$, with either $b' := x \vee b$ or $a' := a \wedge x$, and thus having either $a' \nleq b$ or $a \nleq b'$
The goal is to show that A lattice is distributive if and only if it has the anti-blocking property.
Left to right implication follows easily by assuming is not true and using the distributive laws to get a contradiction. However, the other direction seems a little bit harder. I have proved it (I think), but the proof is not so simple.
My questions is the following then, how can we prove this, that every lattice with the anti-blocking property is distributive? Maybe some of you know a simple proof of this fact, or it is interested in finding one. Thanks anyway!.
PD: Should I show my proof also? If so, should I post that as an answer, or edit this?
edit: I did not mentioned it, but I'm thinking on a proof not using (directly) the fact that a lattice is distributive iff does not have M3 or N5 as a sublattice.