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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 7500), 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 7500 | . . 3 ⊢ Ord On | |
2 | ordirr 6211 | . . 3 ⊢ (Ord On → ¬ On ∈ On) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ¬ On ∈ On |
4 | elong 6201 | . . 3 ⊢ (On ∈ V → (On ∈ On ↔ Ord On)) | |
5 | 1, 4 | mpbiri 260 | . 2 ⊢ (On ∈ V → On ∈ On) |
6 | 3, 5 | mto 199 | 1 ⊢ ¬ On ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2114 Vcvv 3496 Ord word 6192 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: ordeleqon 7505 ssonprc 7509 sucon 7525 orduninsuc 7560 omelon2 7594 tfr2b 8034 tz7.48-3 8082 infensuc 8697 zorn2lem4 9923 noprc 33251 |
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