# Metamath Proof Explorer

## Theorem ordom

Description: Omega is ordinal. Theorem 7.32 of TakeutiZaring p. 43. (Contributed by NM, 18-Oct-1995) (Proof shortened by Andrew Salmon, 27-Aug-2011)

Ref Expression
Assertion ordom Ord ω

### Proof

Step Hyp Ref Expression
1 dftr2 ( Tr ω ↔ ∀ 𝑦𝑥 ( ( 𝑦𝑥𝑥 ∈ ω ) → 𝑦 ∈ ω ) )
2 onelon ( ( 𝑥 ∈ On ∧ 𝑦𝑥 ) → 𝑦 ∈ On )
3 2 expcom ( 𝑦𝑥 → ( 𝑥 ∈ On → 𝑦 ∈ On ) )
4 limord ( Lim 𝑧 → Ord 𝑧 )
5 ordtr ( Ord 𝑧 → Tr 𝑧 )
6 trel ( Tr 𝑧 → ( ( 𝑦𝑥𝑥𝑧 ) → 𝑦𝑧 ) )
7 4 5 6 3syl ( Lim 𝑧 → ( ( 𝑦𝑥𝑥𝑧 ) → 𝑦𝑧 ) )
8 7 expd ( Lim 𝑧 → ( 𝑦𝑥 → ( 𝑥𝑧𝑦𝑧 ) ) )
9 8 com12 ( 𝑦𝑥 → ( Lim 𝑧 → ( 𝑥𝑧𝑦𝑧 ) ) )
10 9 a2d ( 𝑦𝑥 → ( ( Lim 𝑧𝑥𝑧 ) → ( Lim 𝑧𝑦𝑧 ) ) )
11 10 alimdv ( 𝑦𝑥 → ( ∀ 𝑧 ( Lim 𝑧𝑥𝑧 ) → ∀ 𝑧 ( Lim 𝑧𝑦𝑧 ) ) )
12 3 11 anim12d ( 𝑦𝑥 → ( ( 𝑥 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧𝑥𝑧 ) ) → ( 𝑦 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧𝑦𝑧 ) ) ) )
13 elom ( 𝑥 ∈ ω ↔ ( 𝑥 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧𝑥𝑧 ) ) )
14 elom ( 𝑦 ∈ ω ↔ ( 𝑦 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧𝑦𝑧 ) ) )
15 12 13 14 3imtr4g ( 𝑦𝑥 → ( 𝑥 ∈ ω → 𝑦 ∈ ω ) )
16 15 imp ( ( 𝑦𝑥𝑥 ∈ ω ) → 𝑦 ∈ ω )
17 16 ax-gen 𝑥 ( ( 𝑦𝑥𝑥 ∈ ω ) → 𝑦 ∈ ω )
18 1 17 mpgbir Tr ω
19 omsson ω ⊆ On
20 ordon Ord On
21 trssord ( ( Tr ω ∧ ω ⊆ On ∧ Ord On ) → Ord ω )
22 18 19 20 21 mp3an Ord ω