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

Theorem ordzsl 7866
Description: An ordinal is zero, a successor ordinal, or a limit ordinal. Remark 1.12 of [Schloeder] p. 2. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
ordzsl (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ordzsl
StepHypRef Expression
1 orduninsuc 7864 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
21biimprd 248 . . . . 5 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 = 𝐴))
3 unizlim 6507 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
42, 3sylibd 239 . . . 4 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
54orrd 864 . . 3 (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6 3orass 1090 . . . 4 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7 or12 921 . . . 4 ((𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
86, 7bitri 275 . . 3 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
95, 8sylibr 234 . 2 (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10 ord0 6437 . . . 4 Ord ∅
11 ordeq 6391 . . . 4 (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
1210, 11mpbiri 258 . . 3 (𝐴 = ∅ → Ord 𝐴)
13 onsuc 7831 . . . . . 6 (𝑥 ∈ On → suc 𝑥 ∈ On)
14 eleq1 2829 . . . . . 6 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1513, 14imbitrrid 246 . . . . 5 (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16 eloni 6394 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
1715, 16syl6com 37 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
1817rexlimiv 3148 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19 limord 6444 . . 3 (Lim 𝐴 → Ord 𝐴)
2012, 18, 193jaoi 1430 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
219, 20impbii 209 1 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wo 848  w3o 1086   = wceq 1540  wcel 2108  wrex 3070  c0 4333   cuni 4907  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390
This theorem is referenced by:  onzsl  7867  tfrlem16  8433  omeulem1  8620  oaabs2  8687  rankxplim3  9921  rankxpsuc  9922  cardlim  10012  cardaleph  10129  cflim2  10303  dfrdg2  35796
  Copyright terms: Public domain W3C validator