Theorem onprc 2240

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 of [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinals is ordinal (ordon 2238), it must be both an element of the set of all ordinals 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 ∈ V

Proof of Theorem onprc

Step	Hyp	Ref	Expression
1		ordon 2238	. . 3 ⊢ Ord On
2		ordeirr 2217	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 6	. 2 ⊢ ¬ On ∈ On
4		elong 2207	. . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On))
5	1, 4	mpbiri 169	. 2 ⊢ (On ∈ V → On ∈ On)
6	3, 5	mto 93	1 ⊢ ¬ On ∈ V

Syntax hints:  ¬ wn 1   ∈ wcel 1092  Vcvv 1348  Ord word 2198  Oncon0 2199

This theorem is referenced by:  ordeleqon 2241  sucon 2298  ordunisuc 2339  orduninsuc 2365  tz7.48-3 2996  abianfp 3000  omelon 3476  zornlem4 3606

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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