Theorem onprc 4552

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 4486), 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 4486	. . 3 |- Ord On
2		ordirr 4542	. . 3 |- ( Ord On -> -. On e. On )
3	1, 2	ax-mp 5	. 2 |- -. On e. On
4		elong 4374	. . 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 662	1 |- -. On e. _V

Syntax hints:   -. wn 3   e. wcel 2148   _Vcvv 2738   Ord word 4363   Oncon0 4364

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-sn 3599  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369

This theorem is referenced by:  sucon  4553