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Theorem onprc 4334
Description: No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. 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 4269), 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 4269 . . 3 Ord On
2 ordirr 4324 . . 3 (Ord On → ¬ On ∈ On)
31, 2ax-mp 7 . 2 ¬ On ∈ On
4 elong 4167 . . 3 (On ∈ V → (On ∈ On ↔ Ord On))
51, 4mpbiri 166 . 2 (On ∈ V → On ∈ On)
63, 5mto 621 1 ¬ On ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1436  Vcvv 2614  Ord word 4156  Oncon0 4157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-setind 4319
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-sn 3431  df-uni 3631  df-tr 3905  df-iord 4160  df-on 4162
This theorem is referenced by:  sucon  4335
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