MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordon Structured version   Unicode version

Theorem ordon 4765
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 4606 . 2  |-  Tr  On
2 onfr 4622 . . 3  |-  _E  Fr  On
3 eloni 4593 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4593 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4615 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4499 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 229 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4499 . . . . . . 7  |-  ( y  _E  x  <->  y  e.  x )
96, 7, 83orbi123i 1144 . . . . . 6  |-  ( ( x  _E  y  \/  x  =  y  \/  y  _E  x )  <-> 
( x  e.  y  \/  x  =  y  \/  y  e.  x
) )
105, 9sylibr 205 . . . . 5  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  _E  y  \/  x  =  y  \/  y  _E  x ) )
113, 4, 10syl2an 465 . . . 4  |-  ( ( x  e.  On  /\  y  e.  On )  ->  ( x  _E  y  \/  x  =  y  \/  y  _E  x
) )
1211rgen2a 2774 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4764 . . 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 888 . 2  |-  _E  We  On
15 df-ord 4586 . 2  |-  ( Ord 
On 
<->  ( Tr  On  /\  _E  We  On ) )
161, 14, 15mpbir2an 888 1  |-  Ord  On
Colors of variables: wff set class
Syntax hints:    /\ wa 360    \/ w3o 936    e. wcel 1726   A.wral 2707   class class class wbr 4214   Tr wtr 4304    _E cep 4494    Fr wfr 4540    We wwe 4542   Ord word 4582   Oncon0 4583
This theorem is referenced by:  epweon  4766  onprc  4767  ssorduni  4768  ordeleqon  4771  ordsson  4772  onint  4777  suceloni  4795  limon  4818  tfi  4835  ordom  4856  ordtypelem2  7490  hartogs  7515  card2on  7524  tskwe  7839  alephsmo  7985  ondomon  8440  dford3lem2  27100  dford3  27101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
  Copyright terms: Public domain W3C validator