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Theorem ordom 7071
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.)
Assertion
Ref Expression
ordom Ord ω

Proof of Theorem ordom
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4752 . . 3 (Tr ω ↔ ∀𝑦𝑥((𝑦𝑥𝑥 ∈ ω) → 𝑦 ∈ ω))
2 onelon 5746 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
32expcom 451 . . . . . . 7 (𝑦𝑥 → (𝑥 ∈ On → 𝑦 ∈ On))
4 limord 5782 . . . . . . . . . . . 12 (Lim 𝑧 → Ord 𝑧)
5 ordtr 5735 . . . . . . . . . . . 12 (Ord 𝑧 → Tr 𝑧)
6 trel 4757 . . . . . . . . . . . 12 (Tr 𝑧 → ((𝑦𝑥𝑥𝑧) → 𝑦𝑧))
74, 5, 63syl 18 . . . . . . . . . . 11 (Lim 𝑧 → ((𝑦𝑥𝑥𝑧) → 𝑦𝑧))
87expd 452 . . . . . . . . . 10 (Lim 𝑧 → (𝑦𝑥 → (𝑥𝑧𝑦𝑧)))
98com12 32 . . . . . . . . 9 (𝑦𝑥 → (Lim 𝑧 → (𝑥𝑧𝑦𝑧)))
109a2d 29 . . . . . . . 8 (𝑦𝑥 → ((Lim 𝑧𝑥𝑧) → (Lim 𝑧𝑦𝑧)))
1110alimdv 1844 . . . . . . 7 (𝑦𝑥 → (∀𝑧(Lim 𝑧𝑥𝑧) → ∀𝑧(Lim 𝑧𝑦𝑧)))
123, 11anim12d 586 . . . . . 6 (𝑦𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧𝑥𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧𝑦𝑧))))
13 elom 7065 . . . . . 6 (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧𝑥𝑧)))
14 elom 7065 . . . . . 6 (𝑦 ∈ ω ↔ (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧𝑦𝑧)))
1512, 13, 143imtr4g 285 . . . . 5 (𝑦𝑥 → (𝑥 ∈ ω → 𝑦 ∈ ω))
1615imp 445 . . . 4 ((𝑦𝑥𝑥 ∈ ω) → 𝑦 ∈ ω)
1716ax-gen 1721 . . 3 𝑥((𝑦𝑥𝑥 ∈ ω) → 𝑦 ∈ ω)
181, 17mpgbir 1725 . 2 Tr ω
19 omsson 7066 . 2 ω ⊆ On
20 ordon 6979 . 2 Ord On
21 trssord 5738 . 2 ((Tr ω ∧ ω ⊆ On ∧ Ord On) → Ord ω)
2218, 19, 20, 21mp3an 1423 1 Ord ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1480  wcel 1989  wss 3572  Tr wtr 4750  Ord word 5720  Oncon0 5721  Lim wlim 5722  ωcom 7062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-tr 4751  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-om 7063
This theorem is referenced by:  elnn  7072  omon  7073  limom  7077  ssnlim  7080  omsinds  7081  peano5  7086  nnarcl  7693  nnawordex  7714  oaabslem  7720  oaabs2  7722  omabslem  7723  onomeneq  8147  ominf  8169  findcard3  8200  nnsdomg  8216  dffi3  8334  wofib  8447  alephgeom  8902  iscard3  8913  iunfictbso  8934  unctb  9024  ackbij2lem1  9038  ackbij1lem3  9041  ackbij1lem18  9056  ackbij2  9062  cflim2  9082  fin23lem26  9144  fin23lem23  9145  fin23lem27  9147  fin67  9214  alephexp1  9398  pwfseqlem3  9479  pwcdandom  9486  winainflem  9512  wunex2  9557  om2uzoi  12749  ltweuz  12755  fz1isolem  13240  mreexexdOLD  16303  1stcrestlem  21249  hfuni  32275  hfninf  32277  finxpreclem4  33211
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