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Theorem ordom 4458
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 4457 . . . 4 ((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
21gen2 1394 . . 3 𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
3 dftr2 3968 . . 3 (Tr ω ↔ ∀𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω))
42, 3mpbir 145 . 2 Tr ω
5 treq 3972 . . . 4 (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅))
6 treq 3972 . . . 4 (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥))
7 treq 3972 . . . 4 (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥))
8 tr0 3977 . . . 4 Tr ∅
9 suctr 4281 . . . . 5 (Tr 𝑥 → Tr suc 𝑥)
109a1i 9 . . . 4 (𝑥 ∈ ω → (Tr 𝑥 → Tr suc 𝑥))
115, 6, 7, 6, 8, 10finds 4452 . . 3 (𝑥 ∈ ω → Tr 𝑥)
1211rgen 2444 . 2 𝑥 ∈ ω Tr 𝑥
13 dford3 4227 . 2 (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥))
144, 12, 13mpbir2an 894 1 Ord ω
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1297  wcel 1448  wral 2375  c0 3310  Tr wtr 3966  Ord word 4222  suc csuc 4225  ωcom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-tr 3967  df-iord 4226  df-suc 4231  df-iom 4443
This theorem is referenced by:  omelon2  4459  limom  4465  freccllem  6229  frecfcllem  6231  frecsuclem  6233  fict  6691  infnfi  6718  isinfinf  6720  hashinfuni  10364  hashinfom  10365  hashennn  10367  ennnfonelemrn  11724  ctinf  11735
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