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Theorem ordon 4574
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 4415 . 2  |-  Tr  On
2 onfr 4431 . . 3  |-  _E  Fr  On
3 eloni 4402 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4402 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4424 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4308 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 227 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4308 . . . . . . 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 2609 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4573 . . 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 4395 . 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 1684   A.wral 2543   class class class wbr 4023   Tr wtr 4113    _E cep 4303    Fr wfr 4349    We wwe 4351   Ord word 4391   Oncon0 4392
This theorem is referenced by:  epweon  4575  onprc  4576  ssorduni  4577  ordeleqon  4580  ordsson  4581  onint  4586  suceloni  4604  limon  4627  tfi  4644  ordom  4665  ordtypelem2  7234  hartogs  7259  card2on  7268  tskwe  7583  alephsmo  7729  ondomon  8185  tartarmap  25888  dford3lem2  27120  dford3  27121
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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