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Theorem dfom2 6936
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 inner class builder of non-limit ordinal numbers (see nlimon 6920). (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 6935 . 2 ω = {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)}
2 onsssuc 5716 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥𝑧 ∈ suc 𝑥))
3 ontri1 5660 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥 ↔ ¬ 𝑥𝑧))
42, 3bitr3d 268 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
54ancoms 467 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
6 limeq 5638 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (Lim 𝑦 ↔ Lim 𝑧))
76notbid 306 . . . . . . . . . . 11 (𝑦 = 𝑧 → (¬ Lim 𝑦 ↔ ¬ Lim 𝑧))
87elrab 3330 . . . . . . . . . 10 (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧))
98a1i 11 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
105, 9imbi12d 332 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → ((𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
1110pm5.74da 718 . . . . . . 7 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧)))))
12 vex 3175 . . . . . . . . . . 11 𝑧 ∈ V
13 limelon 5691 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
1412, 13mpan 701 . . . . . . . . . 10 (Lim 𝑧𝑧 ∈ On)
1514pm4.71ri 662 . . . . . . . . 9 (Lim 𝑧 ↔ (𝑧 ∈ On ∧ Lim 𝑧))
1615imbi1i 337 . . . . . . . 8 ((Lim 𝑧𝑥𝑧) ↔ ((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧))
17 impexp 460 . . . . . . . 8 (((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧) ↔ (𝑧 ∈ On → (Lim 𝑧𝑥𝑧)))
18 con34b 304 . . . . . . . . . 10 ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → ¬ Lim 𝑧))
19 ibar 523 . . . . . . . . . . 11 (𝑧 ∈ On → (¬ Lim 𝑧 ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
2019imbi2d 328 . . . . . . . . . 10 (𝑧 ∈ On → ((¬ 𝑥𝑧 → ¬ Lim 𝑧) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2118, 20syl5bb 270 . . . . . . . . 9 (𝑧 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2221pm5.74i 258 . . . . . . . 8 ((𝑧 ∈ On → (Lim 𝑧𝑥𝑧)) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2316, 17, 223bitri 284 . . . . . . 7 ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2411, 23syl6rbbr 277 . . . . . 6 (𝑥 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))))
25 impexp 460 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
26 simpr 475 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ suc 𝑥)
27 suceloni 6882 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
28 onelon 5651 . . . . . . . . . . . 12 ((suc 𝑥 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ On)
2928ex 448 . . . . . . . . . . 11 (suc 𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3027, 29syl 17 . . . . . . . . . 10 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3130ancrd 574 . . . . . . . . 9 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → (𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥)))
3226, 31impbid2 214 . . . . . . . 8 (𝑥 ∈ On → ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) ↔ 𝑧 ∈ suc 𝑥))
3332imbi1d 329 . . . . . . 7 (𝑥 ∈ On → (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3425, 33syl5bbr 272 . . . . . 6 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3524, 34bitrd 266 . . . . 5 (𝑥 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3635albidv 1835 . . . 4 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
37 dfss2 3556 . . . 4 (suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
3836, 37syl6bbr 276 . . 3 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
3938rabbiia 3160 . 2 {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)} = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
401, 39eqtri 2631 1 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172  wss 3539  Oncon0 5626  Lim wlim 5627  suc csuc 5628  ωcom 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-om 6935
This theorem is referenced by:  omsson  6938
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