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Theorem onzsl 4767
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 2908 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
2 eloni 4533 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordzsl 4766 . . . 4  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
4 3mix1 1126 . . . . . 6  |-  ( A  =  (/)  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
54adantl 453 . . . . 5  |-  ( ( A  e.  _V  /\  A  =  (/) )  -> 
( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\ 
Lim  A ) ) )
6 3mix2 1127 . . . . . 6  |-  ( E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
76adantl 453 . . . . 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 1128 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
95, 7, 83jaodan 1250 . . . 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 462 . . 3  |-  ( ( A  e.  _V  /\  Ord  A )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
111, 2, 10syl2anc 643 . 2  |-  ( A  e.  On  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
12 0elon 4576 . . . 4  |-  (/)  e.  On
13 eleq1 2448 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  On  <->  (/)  e.  On ) )
1412, 13mpbiri 225 . . 3  |-  ( A  =  (/)  ->  A  e.  On )
15 suceloni 4734 . . . . 5  |-  ( x  e.  On  ->  suc  x  e.  On )
16 eleq1 2448 . . . . 5  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1715, 16syl5ibrcom 214 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  A  e.  On )
)
1817rexlimiv 2768 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  A  e.  On )
19 limelon 4586 . . 3  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
2014, 18, 193jaoi 1247 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) )  ->  A  e.  On )
2111, 20impbii 181 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 177    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717   E.wrex 2651   _Vcvv 2900   (/)c0 3572   Ord word 4522   Oncon0 4523   Lim wlim 4524   suc csuc 4525
This theorem is referenced by:  oawordeulem  6734  r1pwss  7644  r1val1  7646  pwcfsdom  8392  winalim2  8505  rankcf  8586  dfrdg4  25514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-tr 4245  df-eprel 4436  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529
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