Theorem onprc 4697

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 4631), 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 4631	. . 3 |- Ord On
2		ordirr 4687	. . 3 |- ( Ord On -> -. On e. On )
3	1, 2	ax-mp 5	. 2 |- -. On e. On
4		elong 4516	. . 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 672	1 |- -. On e. _V

Syntax hints:   -. wn 3   e. wcel 2209   _Vcvv 2821   Ord word 4505   Oncon0 4506

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-sn 3714  df-uni 3934  df-tr 4228  df-iord 4509  df-on 4511

This theorem is referenced by:  sucon  4698