▁ ▁which ▁no ▁set ▁has . ▁For ▁example , ▁if ▁ ▁is ▁any ▁existing ▁set , ▁the ▁empty ▁set ▁can ▁be ▁constructed ▁as ▁▁ ▁. ▁ ▁Thus ▁the ▁ax iom ▁of ▁the ▁empty ▁set ▁is ▁imp lied ▁by ▁the ▁nine ▁ax i oms ▁presented ▁here . ▁ ▁The ▁ax iom ▁of ▁ext ensional ity ▁implies ▁the ▁empty ▁set ▁is ▁unique ▁( does ▁not ▁depend ▁on ▁). ▁It ▁is ▁common ▁to ▁make ▁a ▁defin itional ▁extension ▁that ▁adds ▁the ▁symbol ▁"" ▁to ▁the ▁language ▁of ▁Z FC . ▁▁ 4 . ▁A xi om ▁of ▁pair ing ▁▁ ▁If ▁ ▁and ▁ ▁are ▁sets , ▁then ▁there ▁exists ▁a ▁set ▁which ▁contains ▁ ▁and ▁ ▁as ▁elements . ▁▁▁▁ ▁The ▁ax iom ▁schema ▁of ▁specification ▁must ▁be ▁used ▁to ▁reduce ▁this ▁to ▁a ▁set ▁with ▁exactly ▁these ▁two ▁elements . ▁The ▁ax iom ▁of ▁pair ing ▁is ▁part ▁of ▁Z , ▁but ▁is ▁redundant ▁in ▁Z F ▁because ▁it ▁follows ▁from ▁the ▁ax iom ▁schema ▁of ▁replacement , ▁if ▁we ▁are ▁given ▁a ▁set ▁with ▁at ▁least ▁two ▁elements . ▁The ▁existence ▁of ▁a ▁set ▁with ▁at ▁least ▁two ▁elements ▁is ▁assured ▁by ▁either ▁the ▁ax iom ▁of ▁infinity , ▁or ▁by ▁the ▁ax iom ▁schema ▁of ▁specification ▁and ▁ ▁the ▁ax iom ▁of ▁the ▁power ▁set ▁applied ▁twice ▁to ▁any ▁set . ▁▁ 5 . ▁A xi om ▁of ▁union ▁▁ ▁The ▁union ▁over ▁the ▁elements ▁of ▁a ▁set ▁exists . ▁For ▁example , ▁the ▁union ▁over ▁the ▁elements ▁of ▁the ▁set ▁