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Theorem onprc 7141
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 7139), 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
StepHypRef Expression
1 ordon 7139 . . 3 Ord On
2 ordirr 5894 . . 3 (Ord On → ¬ On ∈ On)
31, 2ax-mp 5 . 2 ¬ On ∈ On
4 elong 5884 . . 3 (On ∈ V → (On ∈ On ↔ Ord On))
51, 4mpbiri 248 . 2 (On ∈ V → On ∈ On)
63, 5mto 188 1 ¬ On ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2131  Vcvv 3332  Ord word 5875  Oncon0 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-ord 5879  df-on 5880
This theorem is referenced by:  ordeleqon  7145  ssonprc  7149  sucon  7165  orduninsuc  7200  omelon2  7234  tfr2b  7653  tz7.48-3  7700  infensuc  8295  zorn2lem4  9505  noprc  32193
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