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Theorem onzsl 7553
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 3511 . . 3 (𝐴 ∈ On → 𝐴 ∈ V)
2 eloni 6194 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordzsl 7552 . . . 4 (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
4 3mix1 1324 . . . . . 6 (𝐴 = ∅ → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
54adantl 484 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 = ∅) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
6 3mix2 1325 . . . . . 6 (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
76adantl 484 . . . . 5 ((𝐴 ∈ V ∧ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
8 3mix3 1326 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
95, 7, 83jaodan 1424 . . . 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 586 . 2 (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
12 0elon 6237 . . . 4 ∅ ∈ On
13 eleq1 2898 . . . 4 (𝐴 = ∅ → (𝐴 ∈ On ↔ ∅ ∈ On))
1412, 13mpbiri 260 . . 3 (𝐴 = ∅ → 𝐴 ∈ On)
15 suceloni 7520 . . . . 5 (𝑥 ∈ On → suc 𝑥 ∈ On)
16 eleq1 2898 . . . . 5 (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
1715, 16syl5ibrcom 249 . . . 4 (𝑥 ∈ On → (𝐴 = suc 𝑥𝐴 ∈ On))
1817rexlimiv 3278 . . 3 (∃𝑥 ∈ On 𝐴 = suc 𝑥𝐴 ∈ On)
19 limelon 6247 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
2014, 18, 193jaoi 1421 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → 𝐴 ∈ On)
2111, 20impbii 211 1 (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3o 1080   = wceq 1530  wcel 2107  wrex 3137  Vcvv 3493  c0 4289  Ord word 6183  Oncon0 6184  Lim wlim 6185  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190
This theorem is referenced by:  oawordeulem  8172  r1pwss  9205  r1val1  9207  pwcfsdom  9997  winalim2  10110  rankcf  10191  dfrdg4  33400
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