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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 ⊆ { 𝑥 ∈ On ∣ ¬ Lim 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		omsson	⊢ ω ⊆ On
2		nnlim	⊢ ( 𝑥 ∈ ω → ¬ Lim 𝑥 )
3	2	rgen	⊢ ∀ 𝑥 ∈ ω ¬ Lim 𝑥
4		ssrab	⊢ ( ω ⊆ { 𝑥 ∈ On ∣ ¬ Lim 𝑥 } ↔ ( ω ⊆ On ∧ ∀ 𝑥 ∈ ω ¬ Lim 𝑥 ) )
5	1 3 4	mpbir2an	⊢ ω ⊆ { 𝑥 ∈ On ∣ ¬ Lim 𝑥 }