Theorem onprc 7717

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 7716), 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 7716	. . 3 ⊢ Ord On
2		ordirr 6329	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 5	. 2 ⊢ ¬ On ∈ On
4		elong 6319	. . 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 2113  Vcvv 3437  Ord word 6310  Oncon0 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6314  df-on 6315

This theorem is referenced by:  ordeleqon  7721  ssonprc  7726  sucon  7742  orduninsuc  7779  omelon2  7815  tfr2b  8321  tz7.48-3  8369  infensuc  9075  zorn2lem4  10397  noprc  27720  onvf1od  35172