Axiom of Pairing from Powers and Replacement
From ProofWiki
Theorem
The axiom of pairing is a consequence of:
- the Axiom of Powers
and
- the Axiom of Replacement.
Proof
The set $2 = \left\{\varnothing, \left\{\varnothing\right\}\right\}$ is used with the axiom of replacement as the domain for a mapping whose image is $\left\{A, B \right\}$.
A suitable mapping would be:
- $\left({y = \varnothing \land z = A}\right) \lor \left({y = \left\{{\varnothing}\right\} \land z = B}\right)$
The set $2$ is shown to exist as the set of all subsets of the set of all subsets of the empty set.
Demonstrate that set of all subsets of the set of all subsets of the empty set explicitly.
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$\blacksquare$
