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Theorem onzsl 7867
Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number. (Contributed by NM, 1-Oct-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onzsl (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem onzsl
StepHypRef Expression
1 elex 3501 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 eloni 6394 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordzsl 7866 . . . 4 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
4 3mix1 1331 . . . . . 6 (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
54adantl 481 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
6 3mix2 1332 . . . . . 6 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
76adantl 481 . . . . 5 ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
8 3mix3 1333 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
95, 7, 83jaodan 1433 . . . 4 ((𝐴 ∈ V ∧ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
103, 9sylan2b 594 . . 3 ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
111, 2, 10syl2anc 584 . 2 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
12 0elon 6438 . . . 4 ∅ ∈ On
13 eleq1 2829 . . . 4 (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
1412, 13mpbiri 258 . . 3 (𝐴 = ∅ → 𝐴 ∈ On)
15 onsuc 7831 . . . . 5 (𝑥 ∈ On → suc 𝑥 ∈ On)
16 eleq1 2829 . . . . 5 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1715, 16syl5ibrcom 247 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥𝐴 ∈ On))
1817rexlimiv 3148 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 ∈ On)
19 limelon 6448 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
2014, 18, 193jaoi 1430 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On)
2111, 20impbii 209 1 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3o 1086   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  c0 4333  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:  oawordeulem  8592  r1pwss  9824  r1val1  9826  pwcfsdom  10623  winalim2  10736  rankcf  10817  dfrdg4  35952  naddwordnexlem4  43414
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