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Theorem ordon 4590
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 4431 . 2  |-  Tr  On
2 onfr 4447 . . 3  |-  _E  Fr  On
3 eloni 4418 . . . . 5  |-  ( x  e.  On  ->  Ord  x )
4 eloni 4418 . . . . 5  |-  ( y  e.  On  ->  Ord  y )
5 ordtri3or 4440 . . . . . 6  |-  ( ( Ord  x  /\  Ord  y )  ->  (
x  e.  y  \/  x  =  y  \/  y  e.  x ) )
6 epel 4324 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
7 biid 227 . . . . . . 7  |-  ( x  =  y  <->  x  =  y )
8 epel 4324 . . . . . . 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 2622 . . 3  |-  A. x  e.  On  A. y  e.  On  ( x  _E  y  \/  x  =  y  \/  y  _E  x )
13 dfwe2 4589 . . 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 4411 . 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   Tr wtr 4129    _E cep 4319    Fr wfr 4365    We wwe 4367   Ord word 4407   Oncon0 4408
This theorem is referenced by:  epweon  4591  onprc  4592  ssorduni  4593  ordeleqon  4596  ordsson  4597  onint  4602  suceloni  4620  limon  4643  tfi  4660  ordom  4681  ordtypelem2  7250  hartogs  7275  card2on  7284  tskwe  7599  alephsmo  7745  ondomon  8201  tartarmap  25991  dford3lem2  27223  dford3  27224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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