Lemmas Flashcards
1
Q
A strict total ordering (A, <), iff any non-empty end segment C of A has a <-least element.
A
Proof.
2
Q
Uniqueness theorem for ordered pairs <x,y> = <u,v> iff x=u and y=v
A
Proof.
3
Q
Trans(x) iff US(x)=x
A
Proof.
4
Q
For any x (I) x is a subset of TC(x) and Trans(x); (ii) if Trans(t) and x subset of t, then TC(x) subset of t.; hence TC(x) is the smallest transitive set that satisfies x subset of t. Hence, trans(x) <-> tc(x) = x.
A
Proof.
5
Q
Omega is a set; additionally omega is an inductive set and the smallest one
A
Proof.
6
Q
Omega is transitive
A
Proof.
7
Q
(i) < and <= are transitive; (ii) for all n in omega for any m(m<n <-> S(m) < S(n); (iii) for all m in omega, m doesn’t come before itself
A
Proof.
8
Q
< is a strict total ordering on the natural numbers
A
Proof.
