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Theorem ordon 4703
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 4545 . 2  |-  Tr  On
2 onfr 4561 . . 3  |-  _E  Fr  On
3 eloni 4532 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4532 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4554 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4438 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 228 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4438 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
96, 7, 83orbi123i 1143 . . . . . 6  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
105, 9sylibr 204 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  _E  y  \/  x  =  y  \/  y  _E  x ) )
113, 4, 10syl2an 464 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  _E  y  \/  x  =  y  \/  y  _E  x
) )
1211rgen2a 2715 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4702 . . 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 887 . 2  |-  _E  We  On
15 df-ord 4525 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  _E  We  On ) )
161, 14, 15mpbir2an 887 1  |-  Ord  On
Colors of variables: wff set class
Syntax hints:    /\ wa 359    \/ w3o 935    e. wcel 1717   A.wral 2649   class class class wbr 4153   Tr wtr 4243    _E cep 4433    Fr wfr 4479    We wwe 4481   Ord word 4521   Oncon0 4522
This theorem is referenced by:  epweon  4704  onprc  4705  ssorduni  4706  ordeleqon  4709  ordsson  4710  onint  4715  suceloni  4733  limon  4756  tfi  4773  ordom  4794  ordtypelem2  7421  hartogs  7446  card2on  7455  tskwe  7770  alephsmo  7916  ondomon  8371  dford3lem2  26789  dford3  26790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526
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