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Theorem onzsl 6938
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 3185 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 eloni 5650 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordzsl 6937 . . . 4 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
4 3mix1 1223 . . . . . 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 1224 . . . . . 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 1225 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
95, 7, 83jaodan 1386 . . . 4 ((𝐴 ∈ V ∧ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
103, 9sylan2b 491 . . 3 ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
111, 2, 10syl2anc 691 . 2 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
12 0elon 5695 . . . 4 ∅ ∈ On
13 eleq1 2676 . . . 4 (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
1412, 13mpbiri 247 . . 3 (𝐴 = ∅ → 𝐴 ∈ On)
15 suceloni 6905 . . . . 5 (𝑥 ∈ On → suc 𝑥 ∈ On)
16 eleq1 2676 . . . . 5 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1715, 16syl5ibrcom 236 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥𝐴 ∈ On))
1817rexlimiv 3009 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 ∈ On)
19 limelon 5705 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
2014, 18, 193jaoi 1383 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On)
2111, 20impbii 198 1 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  w3o 1030   = wceq 1475  wcel 1977  wrex 2897  Vcvv 3173  c0 3874  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646
This theorem is referenced by:  oawordeulem  7521  r1pwss  8530  r1val1  8532  pwcfsdom  9284  winalim2  9397  rankcf  9478  dfrdg4  31228
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