Theorem onprc 4619

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 4553), 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 4553	. . 3 |- Ord On
2		ordirr 4609	. . 3 |- ( Ord On -> -. On e. On )
3	1, 2	ax-mp 5	. 2 |- -. On e. On
4		elong 4439	. . 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 664	1 |- -. On e. _V

Syntax hints:   -. wn 3   e. wcel 2178   _Vcvv 2777   Ord word 4428   Oncon0 4429

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4604

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2779  df-dif 3177  df-in 3181  df-ss 3188  df-sn 3650  df-uni 3866  df-tr 4160  df-iord 4432  df-on 4434

This theorem is referenced by:  sucon  4620