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Theorem ordzsl 6679
Description: An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
ordzsl  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Distinct variable group:    x, A

Proof of Theorem ordzsl
StepHypRef Expression
1 orduninsuc 6677 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
21biimprd 223 . . . . 5  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  A  =  U. A ) )
3 unizlim 5003 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
42, 3sylibd 214 . . . 4  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  Lim  A ) ) )
54orrd 378 . . 3  |-  ( Ord 
A  ->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A ) ) )
6 3orass 976 . . . 4  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/  Lim  A ) ) )
7 or12 523 . . . 4  |-  ( ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/ 
Lim  A ) )  <-> 
( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
86, 7bitri 249 . . 3  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
95, 8sylibr 212 . 2  |-  ( Ord 
A  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
10 ord0 4939 . . . 4  |-  Ord  (/)
11 ordeq 4894 . . . 4  |-  ( A  =  (/)  ->  ( Ord 
A  <->  Ord  (/) ) )
1210, 11mpbiri 233 . . 3  |-  ( A  =  (/)  ->  Ord  A
)
13 suceloni 6647 . . . . . 6  |-  ( x  e.  On  ->  suc  x  e.  On )
14 eleq1 2529 . . . . . 6  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1513, 14syl5ibr 221 . . . . 5  |-  ( A  =  suc  x  -> 
( x  e.  On  ->  A  e.  On ) )
16 eloni 4897 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
1715, 16syl6com 35 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  Ord  A ) )
1817rexlimiv 2943 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  Ord  A )
19 limord 4946 . . 3  |-  ( Lim 
A  ->  Ord  A )
2012, 18, 193jaoi 1291 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  ->  Ord  A )
219, 20impbii 188 1  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    \/ w3o 972    = wceq 1395    e. wcel 1819   E.wrex 2808   (/)c0 3793   U.cuni 4251   Ord word 4886   Oncon0 4887   Lim wlim 4888   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893
This theorem is referenced by:  onzsl  6680  tfrlem16  7080  omeulem1  7249  oaabs2  7312  rankxplim3  8316  rankxpsuc  8317  cardlim  8370  cardaleph  8487  cflim2  8660  dfrdg2  29445
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