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Theorem ordon 4677
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 4518 . 2  |-  Tr  On
2 onfr 4534 . . 3  |-  _E  Fr  On
3 eloni 4505 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4505 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4527 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4411 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 227 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4411 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
96, 7, 83orbi123i 1142 . . . . . 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 2694 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4676 . . 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 4498 . 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 934    = wceq 1647    e. wcel 1715   A.wral 2628   class class class wbr 4125   Tr wtr 4215    _E cep 4406    Fr wfr 4452    We wwe 4454   Ord word 4494   Oncon0 4495
This theorem is referenced by:  epweon  4678  onprc  4679  ssorduni  4680  ordeleqon  4683  ordsson  4684  onint  4689  suceloni  4707  limon  4730  tfi  4747  ordom  4768  ordtypelem2  7381  hartogs  7406  card2on  7415  tskwe  7730  alephsmo  7876  ondomon  8332  dford3lem2  26626  dford3  26627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499
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