Theorem onprc 7623

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 7622), 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 7622	. . 3 ⊢ Ord On
2		ordirr 6283	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 5	. 2 ⊢ ¬ On ∈ On
4		elong 6273	. . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On))
5	1, 4	mpbiri 257	. 2 ⊢ (On ∈ V → On ∈ On)
6	3, 5	mto 196	1 ⊢ ¬ On ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2110  Vcvv 3431  Ord word 6264  Oncon0 6265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5197  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-ord 6268  df-on 6269

This theorem is referenced by:  ordeleqon  7627  ssonprc  7632  sucon  7648  orduninsuc  7685  omelon2  7720  tfr2b  8219  tz7.48-3  8267  infensuc  8933  zorn2lem4  10266  noprc  33983