MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onprc Structured version   Visualization version   GIF version

Theorem onprc 7494
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 7493), 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 ∈ V

Proof of Theorem onprc
StepHypRef Expression
1 ordon 7493 . . 3 Ord On
2 ordirr 6197 . . 3 (Ord On → ¬ On ∈ On)
31, 2ax-mp 5 . 2 ¬ On ∈ On
4 elong 6187 . . 3 (On ∈ V → (On ∈ On ↔ Ord On))
51, 4mpbiri 261 . 2 (On ∈ V → On ∈ On)
63, 5mto 200 1 ¬ On ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2115  Vcvv 3481  Ord word 6178  Oncon0 6179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-tr 5160  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-ord 6182  df-on 6183
This theorem is referenced by:  ordeleqon  7498  ssonprc  7502  sucon  7518  orduninsuc  7553  omelon2  7587  tfr2b  8029  tz7.48-3  8077  infensuc  8693  zorn2lem4  9920  noprc  33309
  Copyright terms: Public domain W3C validator