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Theorem onprc 4592
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 4590), 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 4590 . . 3  |-  Ord  On
2 ordirr 4426 . . 3  |-  ( Ord 
On  ->  -.  On  e.  On )
31, 2ax-mp 8 . 2  |-  -.  On  e.  On
4 elong 4416 . . 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 1696   _Vcvv 2801   Ord word 4407   Oncon0 4408
This theorem is referenced by:  ordeleqon  4596  ssonprc  4599  sucon  4615  orduninsuc  4650  omelon2  4684  tfr2b  6428  tz7.48-3  6472  abianfp  6487  infensuc  7055  zorn2lem4  8142  noprc  24406
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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