reserve A,B for limit_ordinal infinite Ordinal;
reserve B1,B2,B3,B5,B6,D, C for Ordinal;
reserve X for set;
reserve X for Subset of A;
reserve M for non countable Aleph;
reserve X for Subset of M;
reserve N,N1 for cardinal infinite Element of M;
reserve A for Ordinal;
reserve x,y,X,Y for set;

theorem Th32:
  (for x st x in X ex y st y in X & x c= y & y is Cardinal)
  implies union X is Cardinal
proof
  assume
A1: for x st x in X ex y st y in X & x c= y & y is Cardinal;
  for x st x in union X holds x is Ordinal & x c= union X
  proof
    let x;
    assume x in union X;
    then consider Y such that
A2: x in Y and
A3: Y in X by TARSKI:def 4;
    consider y such that
A4: y in X and
A5: Y c= y and
A6: y is Cardinal by A1,A3;
    reconsider y1=y as Cardinal by A6;
A7: y1 c= union X by A4,ZFMISC_1:74;
    x in y1 by A2,A5;
    hence x is Ordinal;
    x c= y1 by A2,A5,ORDINAL1:def 2;
    hence thesis by A7;
  end;
  then reconsider UNX = union X as epsilon-transitive epsilon-connected set
by ORDINAL1:19;
A8: UNX c= card UNX
  proof
    let x be object such that
A9: x in UNX;
    reconsider x1=x as Ordinal by A9;
    assume not x in card UNX;
    then card UNX c= x1 by ORDINAL1:16;
    then card UNX in UNX by A9,ORDINAL1:12;
    then consider Y such that
A10: card UNX in Y and
A11: Y in X by TARSKI:def 4;
    consider y such that
A12: y in X and
A13: Y c= y and
A14: y is Cardinal by A1,A11;
    reconsider y1=y as Cardinal by A14;
    card y1 c= card UNX by A12,CARD_1:11,ZFMISC_1:74;
    then
A15: y1 c= card UNX;
    card UNX in y1 by A10,A13;
    then card UNX in card UNX by A15;
    hence contradiction;
  end;
  card UNX c= UNX by CARD_1:8;
  hence thesis by A8,XBOOLE_0:def 10;
end;
