# Show that axiom of replacement implies from the Axiom of Specification

Axiom $3.6$ (Replacement). Let $A$ be a set. For any object $x \in A$ and any
object $y$, suppose we have a statement $P(x, y)$ pertaining to $x$ and $y$,
such that for each $x\in A$ there is at most one $y$ for which $P(x,y)$ is
true. Then there exists a set $\{y: P(x, y) \text{ is true for some } x \in A\}$
such that for апy object $z$, $$z\in \{y : P(x, y)\text{ is true for some } x \in A\} \iff P(x,z)\text{ is true for some } x \in A.$$

Axiom $3.5$ (Specification). Let $A$ be a set, and for each x$\in$ $A$, let $P(x)$ be a property pertaining to $x$ (i.e., $P(x)$ is either a
true statement or a false statement). Then there exists a set, called
$\{x \in A : P(x) \text{ is true}\}$ (or simply $\{x \in A : P(x) \text{ for short}\}$), whose elements are precisely the elements $x$ in $A$ for which $P(x)$ is true. In other words, for any object $y$, $$у \in \{x \in A: P(x)\text{ is true}\} \iff (y \in A \text{ and } P(y)\text{ is true}).$$

I have to show that $3.6\implies 3.5.$

Proof: By $(3.6)$ we can assume the following set $$\{x:P(x,x)\text{ is true for some }x\in A\}.$$ Let $Q(x)=P(x,x)$ then we get the set $$\{x\in A:Q(x) \text{ is true for some }x\},$$
which is what $(3.5)$ wants. Is this proof correct?

PS. I have read other answers to this question on MSE, but none of them use this formulation of the axiom and I guess use more formal notation. I am learning from Tao’s Analysis book and so I’ve not been introduced to such notation.

#### Solutions Collecting From Web of "Show that axiom of replacement implies from the Axiom of Specification"

Not exactly…

From (3.6) :

$y ∈ \{ z \mid \exists x (x \in A \text { and } P(x,z)) \} \Leftrightarrow \text P(x,y) \text { holds for some } x \in A$.

You have to recall that: $y \in \{ z \mid \phi(z) \} \Leftrightarrow \phi(y)$.

Thus the LHS is: $\exists x (x \in A \text { and } P(x,z))$ and it is the same as the RHS.

But the LHS can be rewritten also as :

$y ∈ \{ z \in A \mid \exists x (P(x,z)) \}$

that is : $y ∈ \{ z \in A \mid Q(z) \}$ and again:

$y ∈ \{ z \in A \mid Q(z) \} \Leftrightarrow y \in A \text { and } Q(y) \text { holds }$.