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Theorem ordon 4573
Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Assertion
Ref Expression
ordon  |-  Ord  On

Proof of Theorem ordon
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tron 4414 . 2  |-  Tr  On
2 onfr 4430 . . 3  |-  _E  Fr  On
3 eloni 4401 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4401 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4423 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4307 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 227 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4307 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
96, 7, 83orbi123i 1141 . . . . . 6  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
105, 9sylibr 203 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  _E  y  \/  x  =  y  \/  y  _E  x ) )
113, 4, 10syl2an 463 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  _E  y  \/  x  =  y  \/  y  _E  x
) )
1211rgen2a 2610 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4572 . . 3  |-  (  _E  We  On  <->  (  _E  Fr  On  /\  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x ) ) )
142, 12, 13mpbir2an 886 . 2  |-  _E  We  On
15 df-ord 4394 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  _E  We  On ) )
161, 14, 15mpbir2an 886 1  |-  Ord  On
Colors of variables: wff set class
Syntax hints:    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1685   A.wral 2544   class class class wbr 4024   Tr wtr 4114    _E cep 4302    Fr wfr 4348    We wwe 4350   Ord word 4390   Oncon0 4391
This theorem is referenced by:  epweon  4574  onprc  4575  ssorduni  4576  ordeleqon  4579  ordsson  4580  onint  4585  suceloni  4603  limon  4626  tfi  4643  ordom  4664  ordtypelem2  7230  hartogs  7255  card2on  7264  tskwe  7579  alephsmo  7725  ondomon  8181  tartarmap  25299  dford3lem2  26531  dford3  26532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395
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