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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 7478), 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 7478 | . . 3 ⊢ Ord On | |
2 | ordirr 6177 | . . 3 ⊢ (Ord On → ¬ On ∈ On) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ¬ On ∈ On |
4 | elong 6167 | . . 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 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2111 Vcvv 3441 Ord word 6158 Oncon0 6159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 |
This theorem is referenced by: ordeleqon 7483 ssonprc 7487 sucon 7503 orduninsuc 7538 omelon2 7572 tfr2b 8015 tz7.48-3 8063 infensuc 8679 zorn2lem4 9910 noprc 33362 |
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