Theorem onprc 7264

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 7263), 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 ∈ V

Proof of Theorem onprc

Step	Hyp	Ref	Expression
1		ordon 7263	. . 3 ⊢ Ord On
2		ordirr 5996	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 5	. 2 ⊢ ¬ On ∈ On
4		elong 5986	. . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On))
5	1, 4	mpbiri 250	. 2 ⊢ (On ∈ V → On ∈ On)
6	3, 5	mto 189	1 ⊢ ¬ On ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2107  Vcvv 3398  Ord word 5977  Oncon0 5978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-tr 4990  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-ord 5981  df-on 5982

This theorem is referenced by:  ordeleqon  7268  ssonprc  7272  sucon  7288  orduninsuc  7323  omelon2  7357  tfr2b  7777  tz7.48-3  7824  infensuc  8428  zorn2lem4  9658  noprc  32488