Theorem onprc 2989

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 2987), 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 2987	. . 3 |- Ord On
2		ordirr 2966	. . 3 |- (Ord On -> -. On e. On)
3	1, 2	ax-mp 7	. 2 |- -. On e. On
4		elong 2956	. . 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 958  Vcvv 1811  Ord word 2947  Oncon0 2948

This theorem is referenced by:  ordeleqon 2990  sucon 3045  ordunisuc 3089  orduninsuc 3114  tz7.48-3 3958  abianfp 3962  omelon 4629  zorn2lem4 4791

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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