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Theorem onprc 4576
Description: No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4574), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
Assertion
Ref Expression
onprc  |-  -.  On  e.  _V

Proof of Theorem onprc
StepHypRef Expression
1 ordon 4574 . . 3  |-  Ord  On
2 ordirr 4410 . . 3  |-  ( Ord 
On  ->  -.  On  e.  On )
31, 2ax-mp 8 . 2  |-  -.  On  e.  On
4 elong 4400 . . 3  |-  ( On  e.  _V  ->  ( On  e.  On  <->  Ord  On ) )
51, 4mpbiri 224 . 2  |-  ( On  e.  _V  ->  On  e.  On )
63, 5mto 167 1  |-  -.  On  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684   _Vcvv 2788   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ordeleqon  4580  ssonprc  4583  sucon  4599  orduninsuc  4634  omelon2  4668  tfr2b  6412  tz7.48-3  6456  abianfp  6471  infensuc  7039  zorn2lem4  8126  noprc  24335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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