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Theorem ordon 4576
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 4417 . 2  |-  Tr  On
2 onfr 4433 . . 3  |-  _E  Fr  On
3 eloni 4404 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4404 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4426 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4310 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 227 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4310 . . . . . . 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 2611 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4575 . . 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 4397 . 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 1625    e. wcel 1686   A.wral 2545   class class class wbr 4025   Tr wtr 4115    _E cep 4305    Fr wfr 4351    We wwe 4353   Ord word 4393   Oncon0 4394
This theorem is referenced by:  epweon  4577  onprc  4578  ssorduni  4579  ordeleqon  4582  ordsson  4583  onint  4588  suceloni  4606  limon  4629  tfi  4646  ordom  4667  ordtypelem2  7236  hartogs  7261  card2on  7270  tskwe  7585  alephsmo  7731  ondomon  8187  tartarmap  25899  dford3lem2  27131  dford3  27132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398
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