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Metamath Proof Explorer

Theorem trom

Description: The class of finite ordinals _om is a transitive class. (Contributed by NM, 18-Oct-1995) (Proof shortened by Andrew Salmon, 27-Aug-2011)

Ref Expression
Assertion trom Tr ω

Proof

Step Hyp Ref Expression
1 dftr2 ⊢ ( Tr ω ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω ) → 𝑦 ∈ ω ) )
2 onelon ⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ On )
3 2 expcom ⊢ ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ On → 𝑦 ∈ On ) )
4 limord ⊢ ( Lim 𝑧 → Ord 𝑧 )
5 ordtr ⊢ ( Ord 𝑧 → Tr 𝑧 )
6 trel ⊢ ( Tr 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑧 ) )
7 4 5 6 3syl ⊢ ( Lim 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧 ) → 𝑦 ∈ 𝑧 ) )
8 7 expd ⊢ ( Lim 𝑧 → ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) )
9 8 com12 ⊢ ( 𝑦 ∈ 𝑥 → ( Lim 𝑧 → ( 𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧 ) ) )
10 9 a2d ⊢ ( 𝑦 ∈ 𝑥 → ( ( Lim 𝑧 → 𝑥 ∈ 𝑧 ) → ( Lim 𝑧 → 𝑦 ∈ 𝑧 ) ) )
11 10 alimdv ⊢ ( 𝑦 ∈ 𝑥 → ( ∀ 𝑧 ( Lim 𝑧 → 𝑥 ∈ 𝑧 ) → ∀ 𝑧 ( Lim 𝑧 → 𝑦 ∈ 𝑧 ) ) )
12 3 11 anim12d ⊢ ( 𝑦 ∈ 𝑥 → ( ( 𝑥 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧 → 𝑥 ∈ 𝑧 ) ) → ( 𝑦 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧 → 𝑦 ∈ 𝑧 ) ) ) )
13 elom ⊢ ( 𝑥 ∈ ω ↔ ( 𝑥 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧 → 𝑥 ∈ 𝑧 ) ) )
14 elom ⊢ ( 𝑦 ∈ ω ↔ ( 𝑦 ∈ On ∧ ∀ 𝑧 ( Lim 𝑧 → 𝑦 ∈ 𝑧 ) ) )
15 12 13 14 3imtr4g ⊢ ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ ω → 𝑦 ∈ ω ) )
16 15 imp ⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω ) → 𝑦 ∈ ω )
17 16 ax-gen ⊢ ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω ) → 𝑦 ∈ ω )
18 1 17 mpgbir ⊢ Tr ω