MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onzsl Structured version   Unicode version

Theorem onzsl 4818
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 2956 . . 3  |-  ( A  e.  On  ->  A  e.  _V )
2 eloni 4583 . . 3  |-  ( A  e.  On  ->  Ord  A )
3 ordzsl 4817 . . . 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 4626 . . . 4  |-  (/)  e.  On
13 eleq1 2495 . . . 4  |-  ( A  =  (/)  ->  ( A  e.  On  <->  (/)  e.  On ) )
1412, 13mpbiri 225 . . 3  |-  ( A  =  (/)  ->  A  e.  On )
15 suceloni 4785 . . . . 5  |-  ( x  e.  On  ->  suc  x  e.  On )
16 eleq1 2495 . . . . 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 2816 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  A  e.  On )
19 limelon 4636 . . 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 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   (/)c0 3620   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575
This theorem is referenced by:  oawordeulem  6789  r1pwss  7702  r1val1  7704  pwcfsdom  8450  winalim2  8563  rankcf  8644  dfrdg4  25787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579
  Copyright terms: Public domain W3C validator