Article WELLSET1, MML version 4.99.1005

:: WELLSET1:th 1
theorem
for b1 being set
for b2 being Relation-like set holds
      b1 in field b2
   iff
      ex b3 being set st
         ([b1,b3] in b2 or [b3,b1] in b2);

:: WELLSET1:th 3
theorem
for b1, b2 being set
for b3 being Relation-like set
      st b1 <> {} & b2 <> {} & b3 = [:b1,b2:]
   holds field b3 = b1 \/ b2;

:: WELLSET1:sch 1
scheme WELLSET1:sch 1
{F1 -> set}:
ex b1 being set st
   for b2 being Relation-like set holds
         b2 in b1
      iff
         b2 in F1() & P1[b2]


:: WELLSET1:th 6
theorem
for b1, b2 being set
for b3 being Relation-like set
      st b1 in field b3 & b2 in field b3 & b3 is well-ordering & not b1 in b3 -Seg b2
   holds [b2,b1] in b3;

:: WELLSET1:th 7
theorem
for b1, b2 being set
for b3 being Relation-like set
      st b1 in field b3 & b2 in field b3 & b3 is well-ordering & b1 in b3 -Seg b2
   holds not [b2,b1] in b3;

:: WELLSET1:th 8
theorem
for b1 being Relation-like Function-like set
for b2 being set
      st for b3 being set
              st b3 in b2
           holds not b1 . b3 in b3 & b1 . b3 in union b2
   holds ex b3 being Relation-like set st
      field b3 c= union b2 &
       b3 is well-ordering &
       not field b3 in b2 &
       (for b4 being set
             st b4 in field b3
          holds b3 -Seg b4 in b2 & b1 . (b3 -Seg b4) = b4);

:: WELLSET1:th 9
theorem
for b1 being set holds
   ex b2 being Relation-like set st
      b2 is well-ordering & field b2 = b1;