Theorem onprc 7784

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 7783), 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

Step	Hyp	Ref	Expression
1		ordon 7783	. . 3 ⊢ Ord On
2		ordirr 6393	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 5	. 2 ⊢ ¬ On ∈ On
4		elong 6383	. . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On))
5	1, 4	mpbiri 258	. 2 ⊢ (On ∈ V → On ∈ On)
6	3, 5	mto 197	1 ⊢ ¬ On ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2105  Vcvv 3477  Ord word 6374  Oncon0 6375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5297  ax-nul 5304  ax-pr 5427

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5147  df-opab 5209  df-tr 5264  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5630  df-we 5632  df-ord 6378  df-on 6379

This theorem is referenced by:  ordeleqon  7788  ssonprc  7794  sucon  7810  orduninsuc  7851  omelon2  7887  tfr2b  8422  tz7.48-3  8470  infensuc  9181  zorn2lem4  10523  noprc  27818