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Mirrors > Home > MPE Home > Th. List > onprc | Structured version Visualization version GIF version |
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 7780), 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.) |
Ref | Expression |
---|---|
onprc | ⊢ ¬ On ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 7780 | . . 3 ⊢ Ord On | |
2 | ordirr 6389 | . . 3 ⊢ (Ord On → ¬ On ∈ On) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ¬ On ∈ On |
4 | elong 6379 | . . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On)) | |
5 | 1, 4 | mpbiri 257 | . 2 ⊢ (On ∈ V → On ∈ On) |
6 | 3, 5 | mto 196 | 1 ⊢ ¬ On ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2098 Vcvv 3461 Ord word 6370 Oncon0 6371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-ord 6374 df-on 6375 |
This theorem is referenced by: ordeleqon 7785 ssonprc 7791 sucon 7807 orduninsuc 7848 omelon2 7884 tfr2b 8417 tz7.48-3 8465 infensuc 9180 zorn2lem4 10524 noprc 27758 |
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