Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Question

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if $(a\preceq_A a') \wedge (b\preceq_B b')$. I'm to prove here that $\preccurlyeq$ is a partial order on $C$. I know that to prove it is indeed a partial order, it must be reflexive, transitive and anti-symmetric. So I think for reflexivity you'd have $(a,a) \preccurlyeq$ $(a',a')$ where $C = A \times A$, which is pretty obvious. I'm not sure how one would prove the other parts, though.

Answer 1

$[(a,b) \preccurlyeq$ $(a',b')] \wedge [(a',b') \preccurlyeq$ $(a'',b'')] \iff$ $[(a\preceq_A a') \wedge (b\preceq_B b')] \wedge[(a'\preceq_A a'') \wedge (b'\preceq_B b'')] \iff$ $ [(a\preceq_A a') \wedge (a'\preceq_A a'')] \wedge[(b\preceq_B b') \wedge (b'\preceq_B b'')] \Rightarrow $
$[(a\preceq_A a'') \wedge (b\preceq_B b'')] \iff $ $[(a,b) \preccurlyeq$ $(a'',b'')]$

and so on...