let V be Universe; :: thesis: for X being Subclass of V
for o, p being set st X is closed_wrt_A1-A7 & o in X & p in X holds
( {o,p} in X & [o,p] in X )
let X be Subclass of V; :: thesis: for o, p being set st X is closed_wrt_A1-A7 & o in X & p in X holds
( {o,p} in X & [o,p] in X )
let o, p be set ; :: thesis: ( X is closed_wrt_A1-A7 & o in X & p in X implies ( {o,p} in X & [o,p] in X ) )
assume A1: ( X is closed_wrt_A1-A7 & o in X & p in X ) ; :: thesis: ( {o,p} in X & [o,p] in X )
then A2: X is closed_wrt_A2 by Def13;
reconsider a = o, b = p as Element of V by A1;
A3: {a,b} in X by A1, A2, Def7;
thus {o,p} in X by A1, A2, Def7; :: thesis: [o,p] in X
{o} in X by A1, Th2;
hence [o,p] in X by A2, A3, Def7; :: thesis: verum