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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 6509 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ (𝐴 = ∅ ∨ Lim 𝐴)))
42, 3sylibd 239 . . . 4 (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ Lim 𝐴)))
54orrd 863 . . 3 (Ord 𝐴 → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 = ∅ ∨ Lim 𝐴)))
6 3orass 1089 . . . 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 234 . 2 (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
10 ord0 6439 . . . 4 Ord ∅
11 ordeq 6393 . . . 4 (𝐴 = ∅ → (Ord 𝐴 ↔ Ord ∅))
1210, 11mpbiri 258 . . 3 (𝐴 = ∅ → Ord 𝐴)
13 onsuc 7831 . . . . . 6 (𝑥 ∈ On → suc 𝑥 ∈ On)
14 eleq1 2827 . . . . . 6 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1513, 14imbitrrid 246 . . . . 5 (𝐴 = suc 𝑥 → (𝑥 ∈ On → 𝐴 ∈ On))
16 eloni 6396 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
1715, 16syl6com 37 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥 → Ord 𝐴))
1817rexlimiv 3146 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → Ord 𝐴)
19 limord 6446 . . 3 (Lim 𝐴 → Ord 𝐴)
2012, 18, 193jaoi 1427 . 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 847  w3o 1085   = wceq 1537  wcel 2106  wrex 3068  c0 4339   cuni 4912  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392
This theorem is referenced by:  onzsl  7867  tfrlem16  8432  omeulem1  8619  oaabs2  8686  rankxplim3  9919  rankxpsuc  9920  cardlim  10010  cardaleph  10127  cflim2  10301  dfrdg2  35777
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