Theorem trom 7827

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 5209	. 2 ⊢ (Tr ω ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω))
2		onelon 6350	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
3	2	expcom 413	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ On))
4		limord 6386	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → Ord 𝑧)
5		ordtr 6339	. . . . . . . . . . 11 ⊢ (Ord 𝑧 → Tr 𝑧)
6		trel 5215	. . . . . . . . . . 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 415	. . . . . . . . 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 1918	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) → ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧)))
12	3, 11	anim12d 610	. . . . 5 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))))
13		elom 7821	. . . . 5 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)))
14		elom 7821	. . . . 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 406	. . 3 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)
17	16	ax-gen 1797	. 2 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)
18	1, 17	mpgbir 1801	1 ⊢ Tr ω

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   ∈ wcel 2114  Tr wtr 5207  Ord word 6324  Oncon0 6325  Lim wlim 6326  ωcom 7818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-lim 6330  df-om 7819

This theorem is referenced by:  ordom  7828  elnn  7829  omsinds  7839