Theorem onprc 4656

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 4590), 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 e. _V

Proof of Theorem onprc

Step	Hyp	Ref	Expression
1		ordon 4590	. . 3 |- Ord On
2		ordirr 4646	. . 3 |- ( Ord On -> -. On e. On )
3	1, 2	ax-mp 5	. 2 |- -. On e. On
4		elong 4476	. . 3 |- ( On e. _V -> ( On e. On <-> Ord On ) )
5	1, 4	mpbiri 168	. 2 |- ( On e. _V -> On e. On )
6	3, 5	mto 668	1 |- -. On e. _V

Syntax hints:   -. wn 3   e. wcel 2202   _Vcvv 2803   Ord word 4465   Oncon0 4466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-sn 3679  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471

This theorem is referenced by:  sucon  4657