defpred S₁[ set ] means ex a being Element of R st $1 = Class a;
let F1, F2 be Subset-Family of R; :: thesis: ( ( for A being Subset of R holds
( A in F1 iff ex a being Element of R st A = Class a ) ) & ( for A being Subset of R holds
( A in F2 iff ex a being Element of R st A = Class a ) ) implies F1 = F2 )
assume A1: for A being Subset of R holds
( A in F1 iff S₁[A] ) ; :: thesis: ( ex A being Subset of R st
( ( A in F2 implies ex a being Element of R st A = Class a ) implies ( ex a being Element of R st A = Class a & not A in F2 ) ) or F1 = F2 )
assume A2: for A being Subset of R holds
( A in F2 iff S₁[A] ) ; :: thesis: F1 = F2
thus F1 = F2 from SUBSET_1:sch 4(A1, A2); :: thesis: verum