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Theorem ordom 4376
Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
Assertion
Ref Expression
ordom Ord ω

Proof of Theorem ordom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn 4375 . . . 4 ((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
21gen2 1380 . . 3 𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
3 dftr2 3898 . . 3 (Tr ω ↔ ∀𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω))
42, 3mpbir 144 . 2 Tr ω
5 treq 3902 . . . 4 (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅))
6 treq 3902 . . . 4 (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥))
7 treq 3902 . . . 4 (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥))
8 tr0 3907 . . . 4 Tr ∅
9 suctr 4205 . . . . 5 (Tr 𝑥 → Tr suc 𝑥)
109a1i 9 . . . 4 (𝑥 ∈ ω → (Tr 𝑥 → Tr suc 𝑥))
115, 6, 7, 6, 8, 10finds 4370 . . 3 (𝑥 ∈ ω → Tr 𝑥)
1211rgen 2421 . 2 𝑥 ∈ ω Tr 𝑥
13 dford3 4151 . 2 (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥))
144, 12, 13mpbir2an 884 1 Ord ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283  wcel 1434  wral 2353  c0 3268  Tr wtr 3896  Ord word 4146  suc csuc 4149  ωcom 4360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-uni 3623  df-int 3658  df-tr 3897  df-iord 4150  df-suc 4155  df-iom 4361
This theorem is referenced by:  omelon2  4377  limom  4383  freccllem  6072  frecfcllem  6074  frecsuclem  6076  fict  6425  infnfi  6452  isinfinf  6454  hashinfuni  9837  hashinfom  9838  hashennn  9840
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