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Theorem dfom2 7232
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 7216). (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 7231 . 2 ω = {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)}
2 onsssuc 5974 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥𝑧 ∈ suc 𝑥))
3 ontri1 5918 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥 ↔ ¬ 𝑥𝑧))
42, 3bitr3d 270 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
54ancoms 468 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
6 limeq 5896 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (Lim 𝑦 ↔ Lim 𝑧))
76notbid 307 . . . . . . . . . . 11 (𝑦 = 𝑧 → (¬ Lim 𝑦 ↔ ¬ Lim 𝑧))
87elrab 3504 . . . . . . . . . 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 333 . . . . . . . 8 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → ((𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
1110pm5.74da 725 . . . . . . 7 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧)))))
12 vex 3343 . . . . . . . . . . 11 𝑧 ∈ V
13 limelon 5949 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
1412, 13mpan 708 . . . . . . . . . 10 (Lim 𝑧𝑧 ∈ On)
1514pm4.71ri 668 . . . . . . . . 9 (Lim 𝑧 ↔ (𝑧 ∈ On ∧ Lim 𝑧))
1615imbi1i 338 . . . . . . . 8 ((Lim 𝑧𝑥𝑧) ↔ ((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧))
17 impexp 461 . . . . . . . 8 (((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧) ↔ (𝑧 ∈ On → (Lim 𝑧𝑥𝑧)))
18 con34b 305 . . . . . . . . . 10 ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → ¬ Lim 𝑧))
19 ibar 526 . . . . . . . . . . 11 (𝑧 ∈ On → (¬ Lim 𝑧 ↔ (𝑧 ∈ On ∧ ¬ Lim 𝑧)))
2019imbi2d 329 . . . . . . . . . 10 (𝑧 ∈ On → ((¬ 𝑥𝑧 → ¬ Lim 𝑧) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2118, 20syl5bb 272 . . . . . . . . 9 (𝑧 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2221pm5.74i 260 . . . . . . . 8 ((𝑧 ∈ On → (Lim 𝑧𝑥𝑧)) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2316, 17, 223bitri 286 . . . . . . 7 ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧))))
2411, 23syl6rbbr 279 . . . . . 6 (𝑥 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))))
25 impexp 461 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
26 simpr 479 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ suc 𝑥)
27 suceloni 7178 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
28 onelon 5909 . . . . . . . . . . . 12 ((suc 𝑥 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ On)
2928ex 449 . . . . . . . . . . 11 (suc 𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3027, 29syl 17 . . . . . . . . . 10 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3130ancrd 578 . . . . . . . . 9 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥 → (𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥)))
3226, 31impbid2 216 . . . . . . . 8 (𝑥 ∈ On → ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) ↔ 𝑧 ∈ suc 𝑥))
3332imbi1d 330 . . . . . . 7 (𝑥 ∈ On → (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3425, 33syl5bbr 274 . . . . . 6 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3524, 34bitrd 268 . . . . 5 (𝑥 ∈ On → ((Lim 𝑧𝑥𝑧) ↔ (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
3635albidv 1998 . . . 4 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
37 dfss2 3732 . . . 4 (suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦} ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
3836, 37syl6bbr 278 . . 3 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}))
3938rabbiia 3324 . 2 {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)} = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
401, 39eqtri 2782 1 ω = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  wss 3715  Oncon0 5884  Lim wlim 5885  suc csuc 5886  ωcom 7230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-om 7231
This theorem is referenced by:  omsson  7234
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