I assume ZFC.

Let $U$ a set with the following (1), (2), (3):

1. $\omega\in U$

2. $x\in U\ \Rightarrow x\subset U$

3. $x\in U\ \Rightarrow \mathcal{P}(x)\in U$ (where $\mathcal{P}(x):=${$y| y\subset x$})

Then by these premises, I wish prove that $(4)\Leftrightarrow (4')$ where:

4. if $x\subset U$ and $|x|<|U|$ then $x\in U$ (where $|a|$ is the cardinality of the set $a$)

4') If $f: a\to U$ is function and $a\in U$ then $\bigcup_{s\in a}f(s)\in U$ .

The Book of Monk "Introduction to set theory' claim this equivalence as a exercise.

and the implication $(4)\Rightarrow (4')$ is immediate from the book above..

I tried to prove by induction that $|U|\subset U$, or tried to generalize the Mostowsky theorem for get a inijection $|U|\to U$ which preserves the relation '$\in $' (unsuccessfully). I have see also some posts here about this problem, but I hope exist a (relatively) simple answere..

THen I ask: How to prove $(4')\Rightarrow (4)$ ?