Theorem onprc 3865

Description: No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. 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 3863), 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.

Assertion

Ref	Expression
onprc	|- -. On e. _V

Proof of Theorem onprc

Step	Hyp	Ref	Expression
1		ordon 3863	. . 3 |- Ord On
2		ordirr 3676	. . 3 |- (Ord On -> -. On e. On)
3	1, 2	ax-mp 7	. 2 |- -. On e. On
4		elong 3665	. . 3 |- (On e. _V -> (On e. On <-> Ord On))
5	1, 4	mpbiri 211	. 2 |- (On e. _V -> On e. On)
6	3, 5	mto 121	1 |- -. On e. _V

Syntax hints:  -. wn 2   e. wcel 1300  _Vcvv 2292  Ord word 3656  Oncon0 3657

This theorem is referenced by:  ordeleqon 3866  sucon 3889  ordunisucOLD 3912  orduninsuc 3925  tz7.48-3 5167  abianfp 5171  ordtypelem4 5687  omelon 5736  zorn2lem4 5953  noprc 14018  ordtypelem4OLD 15378

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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