Theorem onprc 4230

Description: No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. 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 4178), 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 4178	. . 3 ⊢ Ord On
2		ordirr 4225	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 7	. 2 ⊢ ¬ On ∈ On
4		elong 4076	. . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On))
5	1, 4	mpbiri 157	. 2 ⊢ (On ∈ V → On ∈ On)
6	3, 5	mto 587	1 ⊢ ¬ On ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 1390  Vcvv 2551  Ord word 4065  Oncon0 4066

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071

This theorem is referenced by:  sucon  4231