I have a question: is D-separation implied by the komologorov axioms?

I've proven that it is in some cases:

Premises:

1)A = A|B :. A|BC ≤ A|C
2)C < C|A
3)C < C|B
4) C|AB < C

proof starts:
1)B|C > B {via premise 3
2)A|BC = A * B * C|AB / (C * B|C) {via premise 1
3)A|BC * C = A * B * C|AB / B|C
4)A|BC * C / A = B * C|AB / B|C
5)B * C|AB / B|C < C|AB {via line 1
6)B * C|AB / B|C < C {via line 5 and premise 4
7)A|BC * C / A < C {via lines 6 and 4
8)A|C = A * C|A / C
9)A|C * C = A * C|A
10)A|C * C / A = C|A
11)C < A|C * C / A {via line 10 and premise 2
12)A|BC * C / A < A|C * C / A {via lines 11 and 7
13)A|BC < A|C
Q.E.D.

Premises:

1) A = A|B :. A|BC ≤ A|C
2) C < C|A
3) C < C|B
4) C|AB = C

proof starts:

1)A|C = A * C|A / C
2)A|BC = A * B * C / (B * C|B) {via premises 1 and 4
3)A|BC = A * C / C|B
4)A * C < A * C|A {via premise 2
5)A * C / C|B < A * C|A / C {via line 4 and premise 3
6)A|BC < A|C {via lines 1, 3, and 5
Q.E.D.

If it is implied by classical probability theory, could someone please refer me to a proof?