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Theorem onzsl 6659
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 3122 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
2 eloni 4888 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordzsl 6658 . . . 4  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
4 3mix1 1165 . . . . . 6  |-  ( A  =  (/)  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
54adantl 466 . . . . 5  |-  ( ( A  e.  _V  /\  A  =  (/) )  -> 
( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\ 
Lim  A ) ) )
6 3mix2 1166 . . . . . 6  |-  ( E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
76adantl 466 . . . . 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 1167 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
95, 7, 83jaodan 1294 . . . 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 475 . . 3  |-  ( ( A  e.  _V  /\  Ord  A )  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
111, 2, 10syl2anc 661 . 2  |-  ( A  e.  On  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) ) )
12 0elon 4931 . . . 4  |-  (/)  e.  On
13 eleq1 2539 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  On  <->  (/)  e.  On ) )
1412, 13mpbiri 233 . . 3  |-  ( A  =  (/)  ->  A  e.  On )
15 suceloni 6626 . . . . 5  |-  ( x  e.  On  ->  suc  x  e.  On )
16 eleq1 2539 . . . . 5  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1715, 16syl5ibrcom 222 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  A  e.  On )
)
1817rexlimiv 2949 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  A  e.  On )
19 limelon 4941 . . 3  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
2014, 18, 193jaoi 1291 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A ) )  ->  A  e.  On )
2111, 20impbii 188 1  |-  ( A  e.  On  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  ( A  e.  _V  /\  Lim  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   (/)c0 3785   Ord word 4877   Oncon0 4878   Lim wlim 4879   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884
This theorem is referenced by:  oawordeulem  7200  r1pwss  8198  r1val1  8200  pwcfsdom  8954  winalim2  9070  rankcf  9151  dfrdg4  29177
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