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Theorem dfom2 7857
Description: An alternate definition of the set of natural numbers ω. Definition 7.28 of [TakeutiZaring] p. 42, who use the symbol KI for the restricted class abstraction of non-limit ordinal numbers (see nlimon 7840). (Contributed by NM, 1-Nov-2004.)
Assertion
Ref Expression
dfom2 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}

Proof of Theorem dfom2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-om 7856 . 2 ω = {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)}
2 vex 3479 . . . . . . . . . . 11 𝑧 ∈ V
3 limelon 6429 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
42, 3mpan 689 . . . . . . . . . 10 (Lim 𝑧𝑧 ∈ On)
54pm4.71ri 562 . . . . . . . . 9 (Lim 𝑧 ↔ (𝑧 ∈ On ∧ Lim 𝑧))
65imbi1i 350 . . . . . . . 8 ((Lim 𝑧𝑥𝑧) ↔ ((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧))
7 impexp 452 . . . . . . . 8 (((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧) ↔ (𝑧 ∈ On → (Lim 𝑧𝑥𝑧)))
8 con34b 316 . . . . . . . . . 10 ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → ¬ Lim 𝑧))
9 ibar 530 . . . . . . . . . . 11 (𝑧 ∈ On → (¬ Lim 𝑧 ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
109imbi2d 341 . . . . . . . . . 10 (𝑧 ∈ On → ((¬ 𝑥𝑧 → ¬ Lim 𝑧) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
118, 10bitrid 283 . . . . . . . . 9 (𝑧 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
1211pm5.74i 271 . . . . . . . 8 ((𝑧 ∈ On → (Lim 𝑧𝑥𝑧)) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
136, 7, 123bitri 297 . . . . . . 7 ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
14 onsssuc 6455 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥𝑧 ∈ suc 𝑥))
15 ontri1 6399 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥 ↔ ¬ 𝑥𝑧))
1614, 15bitr3d 281 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
1716ancoms 460 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
18 limeq 6377 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (Lim 𝑦 ↔ Lim 𝑧))
1918notbid 318 . . . . . . . . . . 11 (𝑦 = 𝑧 → (¬ Lim 𝑦 ↔ ¬ Lim 𝑧))
2019elrab 3684 . . . . . . . . . 10 (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧))
2120a1i 11 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
2217, 21imbi12d 345 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → ((𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2322pm5.74da 803 . . . . . . 7 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧)))))
2413, 23bitr4id 290 . . . . . 6 (𝑥 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))))
25 impexp 452 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
26 simpr 486 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ suc 𝑥)
27 onsuc 7799 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
28 onelon 6390 . . . . . . . . . . . 12 ((suc 𝑥 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ On)
2928ex 414 . . . . . . . . . . 11 (suc 𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3027, 29syl 17 . . . . . . . . . 10 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3130ancrd 553 . . . . . . . . 9 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → (𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥)))
3226, 31impbid2 225 . . . . . . . 8 (𝑥 ∈ On → ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) ↔ 𝑧 ∈ suc 𝑥))
3332imbi1d 342 . . . . . . 7 (𝑥 ∈ On → (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3425, 33bitr3id 285 . . . . . 6 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3524, 34bitrd 279 . . . . 5 (𝑥 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3635albidv 1924 . . . 4 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
37 dfss2 3969 . . . 4 (suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
3836, 37bitr4di 289 . . 3 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
3938rabbiia 3437 . 2 {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)} = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
401, 39eqtri 2761 1 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  wss 3949  Oncon0 6365  Lim wlim 6366  suc csuc 6367  ωcom 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856
This theorem is referenced by: (None)
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