let R be Relation; :: thesis: ( R is well-ordering implies for Y being set st Y c= field R & Y <> {} holds
ex a being object st
( a in Y & ( for b being object st b in Y holds
[a,b] in R ) ) )
assume A1: R is well-ordering ; :: thesis: for Y being set st Y c= field R & Y <> {} holds
ex a being object st
( a in Y & ( for b being object st b in Y holds
[a,b] in R ) )
let Y be set ; :: thesis: ( Y c= field R & Y <> {} implies ex a being object st
( a in Y & ( for b being object st b in Y holds
[a,b] in R ) ) )
assume that
A2: Y c= field R and
A3: Y <> {} ; :: thesis: ex a being object st
( a in Y & ( for b being object st b in Y holds
[a,b] in R ) )
consider a being object such that
A4: a in Y and
A5: R -Seg a misses Y by A1, A2, A3, Def2;
take a ; :: thesis: ( a in Y & ( for b being object st b in Y holds
[a,b] in R ) )
thus a in Y by A4; :: thesis: for b being object st b in Y holds
[a,b] in R
let b be object ; :: thesis: ( b in Y implies [a,b] in R )
assume A6: b in Y ; :: thesis: [a,b] in R
then not b in R -Seg a by A5, XBOOLE_0:3;
then ( a = b or not [b,a] in R ) by Th1;
then ( a <> b implies [a,b] in R ) by A1, A2, A4, A6, Lm4;
hence [a,b] in R by A1, A2, A4, Lm1; :: thesis: verum