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Theorem ordzsl 7828
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 7826 . . . . . 6 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
21biimprd 247 . . . . 5 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 = 𝐴))
3 unizlim 6483 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
42, 3sylibd 238 . . . 4 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
54orrd 862 . . 3 (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6 3orass 1091 . . . 4 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (𝐴 = ∅ ∨ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)))
7 or12 920 . . . 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 233 . 2 (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10 ord0 6413 . . . 4 Ord ∅
11 ordeq 6367 . . . 4 (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
1210, 11mpbiri 258 . . 3 (𝐴 = ∅ → Ord 𝐴)
13 onsuc 7793 . . . . . 6 (𝑥 ∈ On → suc 𝑥 ∈ On)
14 eleq1 2822 . . . . . 6 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1513, 14imbitrrid 245 . . . . 5 (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16 eloni 6370 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
1715, 16syl6com 37 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
1817rexlimiv 3149 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19 limord 6420 . . 3 (Lim 𝐴 → Ord 𝐴)
2012, 18, 193jaoi 1428 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) → Ord 𝐴)
219, 20impbii 208 1 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wo 846  w3o 1087   = wceq 1542  wcel 2107  wrex 3071  c0 4320   cuni 4906  Ord word 6359  Oncon0 6360  Lim wlim 6361  suc csuc 6362
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-11 2155  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425  ax-un 7719
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-tr 5264  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366
This theorem is referenced by:  onzsl  7829  tfrlem16  8387  omeulem1  8577  oaabs2  8643  rankxplim3  9871  rankxpsuc  9872  cardlim  9962  cardaleph  10079  cflim2  10253  dfrdg2  34704
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