Another Alternative
Another axiom which implies the axiom of pairing in the presence of the axiom of empty set is
- .
Using {} for A and x for B, we get {x} for C. Then use {x} for A and y for B, getting {x,y} for C. One may continue in this fashion to build up any finite set. And this could be used to generate all hereditarily finite sets without using the axiom of union.
Read more about this topic: Axiom Of Pairing
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