Theorem trom 7721

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

Assertion

Ref	Expression
trom	⊢ Tr ω

Proof of Theorem trom

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 5193	. 2 ⊢ (Tr ω ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω))
2		onelon 6291	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
3	2	expcom 414	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ On))
4		limord 6325	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → Ord 𝑧)
5		ordtr 6280	. . . . . . . . . . 11 ⊢ (Ord 𝑧 → Tr 𝑧)
6		trel 5198	. . . . . . . . . . 11 ⊢ (Tr 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑧))
7	4, 5, 6	3syl 18	. . . . . . . . . 10 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑧))
8	7	expd 416	. . . . . . . . 9 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
9	8	com12 32	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → (Lim 𝑧 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
10	9	a2d 29	. . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ((Lim 𝑧 → 𝑥 ∈ 𝑧) → (Lim 𝑧 → 𝑦 ∈ 𝑧)))
11	10	alimdv 1919	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) → ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧)))
12	3, 11	anim12d 609	. . . . 5 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))))
13		elom 7715	. . . . 5 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)))
14		elom 7715	. . . . 5 ⊢ (𝑦 ∈ ω ↔ (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧)))
15	12, 13, 14	3imtr4g 296	. . . 4 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ ω → 𝑦 ∈ ω))
16	15	imp 407	. . 3 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)
17	16	ax-gen 1798	. 2 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)
18	1, 17	mpgbir 1802	1 ⊢ Tr ω

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   ∈ wcel 2106  Tr wtr 5191  Ord word 6265  Oncon0 6266  Lim wlim 6267  ωcom 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-om 7713

This theorem is referenced by:  ordom  7722  elnn  7723  omsinds  7733