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Theorem trom 7895
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.
StepHypRef Expression
1 dftr2 5266 . 2 (Tr ω ↔ ∀𝑦𝑥((𝑦𝑥𝑥 ∈ ω) → 𝑦 ∈ ω))
2 onelon 6410 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
32expcom 413 . . . . . 6 (𝑦𝑥 → (𝑥 ∈ On → 𝑦 ∈ On))
4 limord 6445 . . . . . . . . . . 11 (Lim 𝑧 → Ord 𝑧)
5 ordtr 6399 . . . . . . . . . . 11 (Ord 𝑧 → Tr 𝑧)
6 trel 5273 . . . . . . . . . . 11 (Tr 𝑧 → ((𝑦𝑥𝑥𝑧) → 𝑦𝑧))
74, 5, 63syl 18 . . . . . . . . . 10 (Lim 𝑧 → ((𝑦𝑥𝑥𝑧) → 𝑦𝑧))
87expd 415 . . . . . . . . 9 (Lim 𝑧 → (𝑦𝑥 → (𝑥𝑧𝑦𝑧)))
98com12 32 . . . . . . . 8 (𝑦𝑥 → (Lim 𝑧 → (𝑥𝑧𝑦𝑧)))
109a2d 29 . . . . . . 7 (𝑦𝑥 → ((Lim 𝑧𝑥𝑧) → (Lim 𝑧𝑦𝑧)))
1110alimdv 1913 . . . . . 6 (𝑦𝑥 → (∀𝑧(Lim 𝑧𝑥𝑧) → ∀𝑧(Lim 𝑧𝑦𝑧)))
123, 11anim12d 609 . . . . 5 (𝑦𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧𝑥𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧𝑦𝑧))))
13 elom 7889 . . . . 5 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧𝑥𝑧)))
14 elom 7889 . . . . 5 (𝑦 ∈ ω ↔ (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧𝑦𝑧)))
1512, 13, 143imtr4g 296 . . . 4 (𝑦𝑥 → (𝑥 ∈ ω → 𝑦 ∈ ω))
1615imp 406 . . 3 ((𝑦𝑥𝑥 ∈ ω) → 𝑦 ∈ ω)
1716ax-gen 1791 . 2 𝑥((𝑦𝑥𝑥 ∈ ω) → 𝑦 ∈ ω)
181, 17mpgbir 1795 1 Tr ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1534  wcel 2105  Tr wtr 5264  Ord word 6384  Oncon0 6385  Lim wlim 6386  ωcom 7886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-om 7887
This theorem is referenced by:  ordom  7896  elnn  7897  omsinds  7907
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