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Theorem onprc 4547
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 4481), 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 4481 . . 3 Ord On
2 ordirr 4537 . . 3 (Ord On → ¬ On ∈ On)
31, 2ax-mp 5 . 2 ¬ On ∈ On
4 elong 4369 . . 3 (On ∈ V → (On ∈ On ↔ Ord On))
51, 4mpbiri 168 . 2 (On ∈ V → On ∈ On)
63, 5mto 662 1 ¬ On ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2148  Vcvv 2737  Ord word 4358  Oncon0 4359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-sn 3597  df-uni 3808  df-tr 4099  df-iord 4362  df-on 4364
This theorem is referenced by:  sucon  4548
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