Theorem onprc 2979

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 2977), 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 2977	. . 3 |- Ord On
2		ordirr 2956	. . 3 |- (Ord On -> -. On e. On)
3	1, 2	ax-mp 7	. 2 |- -. On e. On
4		elong 2946	. . 3 |- (On e. V -> (On e. On <-> Ord On))
5	1, 4	mpbiri 194	. 2 |- (On e. V -> On e. On)
6	3, 5	mto 106	1 |- -. On e. V

Syntax hints:  -. wn 2   e. wcel 955  Vcvv 1802  Ord word 2937  Oncon0 2938

This theorem is referenced by:  ordeleqon 2980  sucon 3035  ordunisuc 3079  orduninsuc 3104  tz7.48-3 3943  abianfp 3947  omelon 4601  zorn2lem4 4763

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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