Let's say a set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have:   $ |y| \not > |x|$ .

A set $A$ is hyper-singular, if and only if, there exists a set $x$ such that: $|A|=|\bigcup x|$ and  $| A| > |x|$, and for each $y \in x$ we have: $|y| \not > |x| \land |y| \neq |x|$.

A set $A$ is ultra-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $|y| < |x|$.

Finite examples include $3=\bigcup \{\{0\},\{1,2\}\}$ being super-singular, and $4=\bigcup\{ \{0 \} , \{1\}, \{2,3\} \}$ being ultra-singular. Infinite examples of ultra-singular includes the set union of a Russell's sock. 

> Is it equivalent to AC to say that no infinite super-singlular set exists? 

> Which of the known choice principles non-existence of an infinite set of each of the other two kinds of sets is equivalent to?