Theorem onprc 7777

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 7776), 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 7776	. . 3 ⊢ Ord On
2		ordirr 6379	. . 3 ⊢ (Ord On → ¬ On ∈ On)
3	1, 2	ax-mp 5	. 2 ⊢ ¬ On ∈ On
4		elong 6369	. . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On))
5	1, 4	mpbiri 261	. 2 ⊢ (On ∈ V → On ∈ On)
6	3, 5	mto 200	1 ⊢ ¬ On ∈ V

Syntax hints:  ¬ wn 3   ∈ wcel 2149  Vcvv 3463  Ord word 6360  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by:  ordeleqon  7781  ssonprc  7786  sucon  7802  orduninsuc  7839  omelon2  7875  tfr2b  8383  tz7.48-3  8431  infensuc  9143  ordfin  9200  zorn2lem4  10483  noprc  27915  xoromon  35422  fineqvomonb  35455  onvf1od  35490