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Theorem ordon 6591
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 4890 . 2  |-  Tr  On
2 onfr 4906 . . 3  |-  _E  Fr  On
3 eloni 4877 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4877 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4899 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4783 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 236 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4783 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
96, 7, 83orbi123i 1184 . . . . . 6  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
105, 9sylibr 212 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  _E  y  \/  x  =  y  \/  y  _E  x ) )
113, 4, 10syl2an 475 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  _E  y  \/  x  =  y  \/  y  _E  x
) )
1211rgen2a 2881 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 6590 . . 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 918 . 2  |-  _E  We  On
15 df-ord 4870 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  _E  We  On ) )
161, 14, 15mpbir2an 918 1  |-  Ord  On
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    \/ w3o 970    e. wcel 1823   A.wral 2804   class class class wbr 4439   Tr wtr 4532    _E cep 4778    Fr wfr 4824    We wwe 4826   Ord word 4866   Oncon0 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871
This theorem is referenced by:  epweon  6592  onprc  6593  ssorduni  6594  ordeleqon  6597  ordsson  6598  onint  6603  suceloni  6621  limon  6644  tfi  6661  ordom  6682  ordtypelem2  7936  hartogs  7961  card2on  7972  tskwe  8322  alephsmo  8474  ondomon  8929  dford3lem2  31208  dford3  31209
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