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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 3499 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 eloni 6396 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordzsl 7866 . . . 4 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
4 3mix1 1329 . . . . . 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 1330 . . . . . 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 1331 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
95, 7, 83jaodan 1430 . . . 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 6440 . . . 4 ∅ ∈ On
13 eleq1 2827 . . . 4 (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
1412, 13mpbiri 258 . . 3 (𝐴 = ∅ → 𝐴 ∈ On)
15 onsuc 7831 . . . . 5 (𝑥 ∈ On → suc 𝑥 ∈ On)
16 eleq1 2827 . . . . 5 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1715, 16syl5ibrcom 247 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥𝐴 ∈ On))
1817rexlimiv 3146 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 ∈ On)
19 limelon 6450 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
2014, 18, 193jaoi 1427 . 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 1085   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  c0 4339  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:  oawordeulem  8591  r1pwss  9822  r1val1  9824  pwcfsdom  10621  winalim2  10734  rankcf  10815  dfrdg4  35933  naddwordnexlem4  43391
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