Theorem onprc 4648

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 4582), 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 4582	. . 3 |- Ord On
2		ordirr 4638	. . 3 |- ( Ord On -> -. On e. On )
3	1, 2	ax-mp 5	. 2 |- -. On e. On
4		elong 4468	. . 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 666	1 |- -. On e. _V

Syntax hints:   -. wn 3   e. wcel 2200   _Vcvv 2800   Ord word 4457   Oncon0 4458

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-in 3204  df-ss 3211  df-sn 3673  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463

This theorem is referenced by:  sucon  4649