Do I need parenthesis in this equation or is the hierarchy of the symbols enough?

Question

Do I need parenthesis in this equation or is the hierarchy of the symbols enough? On top of that, is the equation true?

If we have an injection from set $A$ to set $B$, and the cardinality of $B$ is $\aleph_0$, then the cardinality of $A$ is less then or equal to $B$ and the cardinality of the union of $A$ and $B$ is $\aleph_0$.

$A \rightarrowtail B \land |B| = \aleph_0 \to |A|\le|B| \land A \cup B = \aleph_0$

$[(A \rightarrowtail B) \land (|B| = \aleph_0)] \to [(|A|\le|B|) \land (A \cup B = \aleph_0)]$

Answer 1

Operator precedence is something we use for two reasons:

1. To make things more readable for humans, and
2. To give computers unambiguous parse orders for formulas.

Point (2) isn't relevant here since this isn't being fed into a computer, and at least personally I find the unparenthesized formula hard to read, so I'd say that it's not helping with (1), even if there is some established precedence order for all the symbols here.

As for the expression itself it's true, and I expect you know why.