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Theorem onprc 4575
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 4573), 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 4573 . . 3  |-  Ord  On
2 ordirr 4409 . . 3  |-  ( Ord 
On  ->  -.  On  e.  On )
31, 2ax-mp 10 . 2  |-  -.  On  e.  On
4 elong 4399 . . 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 1685   _Vcvv 2789   Ord word 4390   Oncon0 4391
This theorem is referenced by:  ordeleqon  4579  ssonprc  4582  sucon  4598  orduninsuc  4633  omelon2  4667  tfr2b  6407  tz7.48-3  6451  abianfp  6466  infensuc  7034  zorn2lem4  8121  noprc  23735
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395
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