Cardinality of two infinite sets and strictly bigger than relation

Suppose $A$ and $B$ are two infinite sets and cardinality of $B$ is strictly greater than cardinality of $A$.

To prove this strictly gretater than relation between two sets, is it sufficient to only show that there exists no surjective function from $A$ to $B$, or in addition to this, one must also show that there exists a surjection from $B$ to $A$, for proof to be valid.

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If we assume the axiom of choice then every two cardinals are comparable; and every surjection admits an inverse injection. This means that if we assume the axiom of choice then if there is no surjection from $A$ onto $B$ then there is an injection from $A$ into $B$, and there is no injection from $B$ into $A$ either.
If the framework you are working in does not assume the axiom of choice then you have to show that there is an injection from $A$ into $B$ and there is no injection from $B$ into $A$. It is not sufficient to do so with surjections, nor it is sufficient to show that there is a surjection from $B$ onto $A$ and there is no injection from $A$ into $B$.