Is the Axiom of Union independent of the rest of ZF? Short version: Is the axiom of union independent of the rest of axioms of ZF?
NO) Tourlakis (2003) says in p. 177 that the axiom of union can be derived from the rest of ZF if an appropriate version of collection axiom[*] is chosen. The quote is:
«Bourbaki (1966b) adopts the axiom of pairing, but adopts collection version (2), and proves both separation and union»
YES) In the other hand, I have read here and here something like "$H_{\kappa}$ is a model for ZF-Union+¬Union", where $\kappa$ was $\beth_\omega$ or a singular cardinal.
Any reference on the subject would be highly appreciated. I apologize in advance if the question is too basic (not a mathematician!). Also, I have googled it and followed some false trails before asking here. Thanks.
[*] The appropriate version of collection is apparently weaker (or equivalent at most) than the collection axiom that he is adopting in his text. I think the statement is:
$(∀x)(∃z)(∀y)({\mathcal P} [x, y] → y ∈ z) → (\forall A)(\exists B)(\forall y)(y\in B\leftrightarrow (\exists x\in A){\mathcal P}[x,y])$
I have translated the notation from III.8.12 and III.2 (obviating any reference to ur-elements).
EDIT: Thank you very much for the answers, they were really helpful.
 A: The axiom is
$$\forall x \ \exists y \ \forall z \ (\varphi(z,x) \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow  \exists x\in X \ \varphi(y,x)) $$
Consider
$$\forall x \ \exists y \ \forall z \ (z\in x \to z\in y) \to \forall X \ \exists Y \ \forall y \ (y \in Y \leftrightarrow  \exists x\in X \ y\in x ) $$
This is an instance of the axiom. Now the left side is true (take $y=x$). The right side expresses the existence of the union of $X$.

The other version of the axiom:
$$\forall x\in X \ \exists y \ \varphi(x,y) \to \ \exists Y \ \forall x\in X \ \exists y\in Y \ \varphi(x,y)$$
A: Let me add a nice remark I just became aware of: In

Greg Oman. On the axiom of union, Arch. Math. Logic, 49 (3), (2010), 283–289. MR2609983 (2011g:03122),

Oman clarifies precisely which unions can be proved to exist in $\mathsf{ZFC}-\mathrm{Union}$: $\bigcup x$ exists iff $\{|y|\colon y\in x\}$ is bounded. In particular, $A\cup B$ exists for any sets $A,B$. 
See this MSE question for details. The proof uses in essential ways both choice and replacement. 
(Curiously, I do not know of an original reference for the fact that $\mathsf{ZFC}-\mathrm{Union}$ does not suffice to prove the existence of infinite unions, the usual argument being the one in the body of the question, and in Andreas's answer. It would be nice to have the reference, so it can be added here.)
A: The usual version of collection is the second one in Kaveh's answer.  It, with the remaining axioms, won't give the axiom of union.  A counterexample is given in the question, except for a slightly unusual definition of $H_\kappa$ for singular $\kappa$; it should be the collection of those sets $x$ such that each member of the transitive closure $TC(\{x\})$ has cardinality $<\kappa$.  (This does not imply that the whole $TC(\{x\})$ has cardinality $<\kappa$, which is the usual meaning of $H_\kappa$.)
The "appropriate version" of collection used by Bourbaki seems appropriate mainly in the sense that the axiom of union has been built in.
