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Theorem onzsl 4653
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 2809 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
2 eloni 4418 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordzsl 4652 . . . 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 4461 . . . 4  |-  (/)  e.  On
13 eleq1 2356 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  On  <->  (/)  e.  On ) )
1412, 13mpbiri 224 . . 3  |-  ( A  =  (/)  ->  A  e.  On )
15 suceloni 4620 . . . . 5  |-  ( x  e.  On  ->  suc  x  e.  On )
16 eleq1 2356 . . . . 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 2674 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  A  e.  On )
19 limelon 4471 . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   (/)c0 3468   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410
This theorem is referenced by:  oawordeulem  6568  r1pwss  7472  r1val1  7474  pwcfsdom  8221  winalim2  8334  rankcf  8415  dfrdg4  24560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414
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