reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;

theorem Th58:
  x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4
  implies card {x1,x2,x3,x4} = 4
proof
  assume x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4;
  then
A1: card {x1,x2,x3} = 3 & not x4 in {x1,x2,x3} by Th57,ENUMSET1:def 1;
  {x1,x2,x3,x4} = {x1,x2,x3} \/ {x4} by ENUMSET1:6;
  hence card {x1,x2,x3,x4} = 3+1 by A1,Th40
    .= 4;
end;
