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 }