MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfom2 Structured version   Visualization version   GIF version

Theorem dfom2 7214
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 7198). (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 7213 . 2 ω = {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)}
2 onsssuc 5956 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥𝑧 ∈ suc 𝑥))
3 ontri1 5900 . . . . . . . . . . 11 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧𝑥 ↔ ¬ 𝑥𝑧))
42, 3bitr3d 270 . . . . . . . . . 10 ((𝑧 ∈ On ∧ 𝑥 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
54ancoms 455 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (𝑧 ∈ suc 𝑥 ↔ ¬ 𝑥𝑧))
6 limeq 5878 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (Lim 𝑦 ↔ Lim 𝑧))
76notbid 307 . . . . . . . . . . 11 (𝑦 = 𝑧 → (¬ Lim 𝑦 ↔ ¬ Lim 𝑧))
87elrab 3515 . . . . . . . . . 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 805 . . . . . . 7 (𝑥 ∈ On → ((𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})) ↔ (𝑧 ∈ On → (¬ 𝑥𝑧 → (𝑧 ∈ On ∧ ¬ Lim 𝑧)))))
12 vex 3354 . . . . . . . . . . 11 𝑧 ∈ V
13 limelon 5931 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ Lim 𝑧) → 𝑧 ∈ On)
1412, 13mpan 670 . . . . . . . . . 10 (Lim 𝑧𝑧 ∈ On)
1514pm4.71ri 550 . . . . . . . . 9 (Lim 𝑧 ↔ (𝑧 ∈ On ∧ Lim 𝑧))
1615imbi1i 338 . . . . . . . 8 ((Lim 𝑧𝑥𝑧) ↔ ((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧))
17 impexp 437 . . . . . . . 8 (((𝑧 ∈ On ∧ Lim 𝑧) → 𝑥𝑧) ↔ (𝑧 ∈ On → (Lim 𝑧𝑥𝑧)))
18 con34b 305 . . . . . . . . . 10 ((Lim 𝑧𝑥𝑧) ↔ (¬ 𝑥𝑧 → ¬ Lim 𝑧))
19 ibar 518 . . . . . . . . . . 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 437 . . . . . . 7 (((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦}) ↔ (𝑧 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
26 simpr 471 . . . . . . . . 9 ((𝑧 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ suc 𝑥)
27 suceloni 7160 . . . . . . . . . . 11 (𝑥 ∈ On → suc 𝑥 ∈ On)
28 onelon 5891 . . . . . . . . . . . 12 ((suc 𝑥 ∈ On ∧ 𝑧 ∈ suc 𝑥) → 𝑧 ∈ On)
2928ex 397 . . . . . . . . . . 11 (suc 𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3027, 29syl 17 . . . . . . . . . 10 (𝑥 ∈ On → (𝑧 ∈ suc 𝑥𝑧 ∈ On))
3130ancrd 541 . . . . . . . . 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 2001 . . . 4 (𝑥 ∈ On → (∀𝑧(Lim 𝑧𝑥𝑧) ↔ ∀𝑧(𝑧 ∈ suc 𝑥𝑧 ∈ {𝑦 ∈ On ∣ ¬ Lim 𝑦})))
37 dfss2 3740 . . . 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 3334 . 2 {𝑥 ∈ On ∣ ∀𝑧(Lim 𝑧𝑥𝑧)} = {𝑥 ∈ On ∣ suc 𝑥 ⊆ {𝑦 ∈ On ∣ ¬ Lim 𝑦}}
401, 39eqtri 2793 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 382  wal 1629   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  wss 3723  Oncon0 5866  Lim wlim 5867  suc csuc 5868  ωcom 7212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-om 7213
This theorem is referenced by:  omsson  7216
  Copyright terms: Public domain W3C validator