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Theorem onzsl 4637
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  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
Distinct variable group:    x, A

Proof of Theorem onzsl
StepHypRef Expression
1 elex 2796 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
2 eloni 4402 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordzsl 4636 . . . 4  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
4 3mix1 1124 . . . . . 6  |-  ( A  =  (/)  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
54adantl 452 . . . . 5  |-  ( ( A  e.  _V  /\  A  =  (/) )  -> 
( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\ 
Lim  A ) ) )
6 3mix2 1125 . . . . . 6  |-  ( E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
76adantl 452 . . . . 5  |-  ( ( A  e.  _V  /\  E. x  e.  On  A  =  suc  x )  -> 
( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\ 
Lim  A ) ) )
8 3mix3 1126 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
95, 7, 83jaodan 1248 . . . 4  |-  ( ( A  e.  _V  /\  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
103, 9sylan2b 461 . . 3  |-  ( ( A  e.  _V  /\  Ord  A )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
111, 2, 10syl2anc 642 . 2  |-  ( A  e.  On  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
12 0elon 4445 . . . 4  |-  (/)  e.  On
13 eleq1 2343 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  On  <->  (/)  e.  On ) )
1412, 13mpbiri 224 . . 3  |-  ( A  =  (/)  ->  A  e.  On )
15 suceloni 4604 . . . . 5  |-  ( x  e.  On  ->  suc  x  e.  On )
16 eleq1 2343 . . . . 5  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1715, 16syl5ibrcom 213 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  A  e.  On )
)
1817rexlimiv 2661 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  A  e.  On )
19 limelon 4455 . . 3  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
2014, 18, 193jaoi 1245 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) )  ->  A  e.  On )
2111, 20impbii 180 1  |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   (/)c0 3455   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  oawordeulem  6552  r1pwss  7456  r1val1  7458  pwcfsdom  8205  winalim2  8318  rankcf  8399  dfrdg4  23899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
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