Metamath Proof Explorer

Theorem omssnlim

Description: The class of natural numbers is a subclass of the class of non-limit ordinal numbers. Exercise 4 of TakeutiZaring p. 42. (Contributed by NM, 2-Nov-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

Ref Expression
Assertion omssnlim 
|- _om C_ { x e. On | -. Lim x }

Proof

Step Hyp Ref Expression
1 omsson 
 |-  _om C_ On
2 nnlim 
 |-  ( x e. _om -> -. Lim x )
3 2 rgen 
 |-  A. x e. _om -. Lim x
4 ssrab 
 |-  ( _om C_ { x e. On | -. Lim x } <-> ( _om C_ On /\ A. x e. _om -. Lim x ) )
5 1 3 4 mpbir2an 
 |-  _om C_ { x e. On | -. Lim x }