Metamath Proof Explorer


Theorem dfom5

Description: _om is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011) (Revised by Mario Carneiro, 2-Feb-2013)

Ref Expression
Assertion dfom5 ω = { 𝑥 ∣ Lim 𝑥 }

Proof

Step Hyp Ref Expression
1 elom3 ( 𝑦 ∈ ω ↔ ∀ 𝑥 ( Lim 𝑥𝑦𝑥 ) )
2 vex 𝑦 ∈ V
3 2 elintab ( 𝑦 { 𝑥 ∣ Lim 𝑥 } ↔ ∀ 𝑥 ( Lim 𝑥𝑦𝑥 ) )
4 1 3 bitr4i ( 𝑦 ∈ ω ↔ 𝑦 { 𝑥 ∣ Lim 𝑥 } )
5 4 eqriv ω = { 𝑥 ∣ Lim 𝑥 }