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Theorem onprc 4534
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 4532), 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 4532 . . 3  |-  Ord  On
2 ordirr 4368 . . 3  |-  ( Ord 
On  ->  -.  On  e.  On )
31, 2ax-mp 10 . 2  |-  -.  On  e.  On
4 elong 4358 . . 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 2757   Ord word 4349   Oncon0 4350
This theorem is referenced by:  ordeleqon  4538  ssonprc  4541  sucon  4557  orduninsuc  4592  omelon2  4626  tfr2b  6366  tz7.48-3  6410  abianfp  6425  infensuc  6993  zorn2lem4  8080  noprc  23689
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-tr 4074  df-eprel 4263  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354
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