Let \\(a,b \in P\\) and subset \\(A = \\{ a \leq b \; and \; b \leq a \\}\\). Then you get two meets namely, \\(a \wedge b = a\\) and \\(b \wedge a = b\\)?

Using reflexivity and transitivity, we can expand this out into a transitivity triangle and get 1. \\(a \leq b \leq a\\) and \\(a \leq a\\) and 2. \\(b \leq a \leq b\\) and \\(b \leq b\\). Taking \\(a \wedge b\\) in Triangle 1 and Triangle 2 gives a and b respectively.

Using reflexivity and transitivity, we can expand this out into a transitivity triangle and get 1. \\(a \leq b \leq a\\) and \\(a \leq a\\) and 2. \\(b \leq a \leq b\\) and \\(b \leq b\\). Taking \\(a \wedge b\\) in Triangle 1 and Triangle 2 gives a and b respectively.