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Theorem ordom 4329
 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 4328 . . . 4 ((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
21gen2 1339 . . 3 𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω)
3 dftr2 3856 . . 3 (Tr ω ↔ ∀𝑥𝑦((𝑥𝑦𝑦 ∈ ω) → 𝑥 ∈ ω))
42, 3mpbir 134 . 2 Tr ω
5 treq 3860 . . . 4 (𝑦 = ∅ → (Tr 𝑦 ↔ Tr ∅))
6 treq 3860 . . . 4 (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥))
7 treq 3860 . . . 4 (𝑦 = suc 𝑥 → (Tr 𝑦 ↔ Tr suc 𝑥))
8 tr0 3865 . . . 4 Tr ∅
9 suctr 4158 . . . . 5 (Tr 𝑥 → Tr suc 𝑥)
109a1i 9 . . . 4 (𝑥 ∈ ω → (Tr 𝑥 → Tr suc 𝑥))
115, 6, 7, 6, 8, 10finds 4323 . . 3 (𝑥 ∈ ω → Tr 𝑥)
1211rgen 2374 . 2 𝑥 ∈ ω Tr 𝑥
13 dford3 4104 . 2 (Ord ω ↔ (Tr ω ∧ ∀𝑥 ∈ ω Tr 𝑥))
144, 12, 13mpbir2an 849 1 Ord ω
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   ∈ wcel 1393  ∀wral 2306  ∅c0 3224  Tr wtr 3854  Ord word 4099  suc csuc 4102  ωcom 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-suc 4108  df-iom 4314 This theorem is referenced by:  omelon2  4330  limom  4336
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