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Theorem onprc 4467
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 4465), 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 4465 . . 3  |-  Ord  On
2 ordirr 4303 . . 3  |-  ( Ord 
On  ->  -.  On  e.  On )
31, 2ax-mp 10 . 2  |-  -.  On  e.  On
4 elong 4293 . . 3  |-  ( On  e.  _V  ->  ( On  e.  On  <->  Ord  On ) )
51, 4mpbiri 226 . 2  |-  ( On  e.  _V  ->  On  e.  On )
63, 5mto 169 1  |-  -.  On  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 5    e. wcel 1621   _Vcvv 2727   Ord word 4284   Oncon0 4285
This theorem is referenced by:  ordeleqon  4471  ssonprc  4474  sucon  4490  orduninsuc  4525  omelon2  4559  tfr2b  6298  tz7.48-3  6342  abianfp  6357  infensuc  6924  zorn2lem4  8010  noprc  23502
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289
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