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	$$⊢\mathrm{\omega }\subseteq \left\{x\in \mathrm{On}|\neg \mathrm{Lim}x\right\}$$

Proof

Step	Hyp	Ref	Expression
1		omsson	$$⊢\mathrm{\omega }\subseteq \mathrm{On}$$
2		nnlim	$$⊢x\in \mathrm{\omega }\to \neg \mathrm{Lim}x$$
3	2	rgen	$$⊢\forall x\in \mathrm{\omega }\neg \mathrm{Lim}x$$
4		ssrab	$$⊢\mathrm{\omega }\subseteq \left\{x\in \mathrm{On}|\neg \mathrm{Lim}x\right\}\leftrightarrow \left(\mathrm{\omega }\subseteq \mathrm{On}\wedge \forall x\in \mathrm{\omega }\neg \mathrm{Lim}x\right)$$
5	1 3 4	mpbir2an	$$⊢\mathrm{\omega }\subseteq \left\{x\in \mathrm{On}|\neg \mathrm{Lim}x\right\}$$