P(A|C)≥P(B|C) and P(A|C')≥P(B|C')

Prove that P(A)≥P(B).

First I started with,

P(A|C)=P(AC)/P(C) and similarly P(B|C)=P(BC)/P(C).

From above,

P(AC)/P(C)≥P(BC)/P(C) and since P(C)≥0,

P(AC)≥P(BC).

Similarly from above,

P(AC')≥P(BC').

Also, P(AC)=P(A)+P(C)-P(AUC) which led to what is below.

As of now I've gotten down to... but got stuck (Doh)

P(A)≥P(B)-P(BUC)+P(AUC)

P(A)≥P(B)-P(BUC')+P(AUC')