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Theorem onzsl 7835
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 3493 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 eloni 6375 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordzsl 7834 . . . 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 483 . . . . 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 483 . . . . 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 1431 . . . 4 ((𝐴 ∈ V ∧ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴)) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
103, 9sylan2b 595 . . 3 ((𝐴 ∈ V ∧ Ord 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
111, 2, 10syl2anc 585 . 2 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
12 0elon 6419 . . . 4 ∅ ∈ On
13 eleq1 2822 . . . 4 (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
1412, 13mpbiri 258 . . 3 (𝐴 = ∅ → 𝐴 ∈ On)
15 onsuc 7799 . . . . 5 (𝑥 ∈ On → suc 𝑥 ∈ On)
16 eleq1 2822 . . . . 5 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1715, 16syl5ibrcom 246 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥𝐴 ∈ On))
1817rexlimiv 3149 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 ∈ On)
19 limelon 6429 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
2014, 18, 193jaoi 1428 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On)
2111, 20impbii 208 1 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3o 1087   = wceq 1542  wcel 2107  wrex 3071  Vcvv 3475  c0 4323  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371
This theorem is referenced by:  oawordeulem  8554  r1pwss  9779  r1val1  9781  pwcfsdom  10578  winalim2  10691  rankcf  10772  dfrdg4  34923  naddwordnexlem4  42152
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