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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	$⊢ ω \subseteq \{x \in On \| \neg Lim x\}$

Proof

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