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To be specific, I'll state some definitions first.

  1. $0=\emptyset$
    $1=\{\emptyset\}=\{0\}$
    $2=\{\emptyset,\{\emptyset\}\}=\{0,1\}$
    ...
    $n=\{0,1,2,....,n-1\}$

  2. For given set $\text A$ and $\text B$,
    $\text B^\text A=\{ f\in P(\text A \times \text B) $ | $ f:\text A \to \text B \}$

  3. Any function $f:n\to \text{A} $ is finite sequence.
    Any function $f:\mathbb N \to \text A$ is infinite sequence.

In terms of above definitions, $$\bigcup_{n\in \mathbb N} A^n $$ is set of every finite sequence on $A$.

I want to prove this set exists. If I prove $J=\{\text A^0,\text A^1, \text A^2, ....\}$ exists, above set exsits by Axiom of union.

But I have no idea how to prove $J$ exists. I know each $\text A^n$ exist. but how can I construct such a big set?
I'm sure that I can't apply Axiom of pair infinitely many times.

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This is exactly what the Axiom of Replacement is for. Once you know $\mathbb N$ exists, the Axiom of Replacement guarantees that $$ \{ A^i \mid i \in \mathbb N \} $$ is a set.

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Either use the Axiom Schema of Replacement, $$J=\{\,A^n\mid n\in\Bbb N\,\} $$ Or note that all $A^n$ are subsets of $P(\Bbb N\times A)$ and use the Axiom Schema of Separation from there.

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