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Mirrors > Home > MPE Home > Th. List > ncanth | Structured version Visualization version GIF version |
Description: Cantor's theorem fails
for the universal class (which is not a set but a
proper class by vprc 5183). Specifically, the identity function maps
the
universe onto its power class. Compare canth 7090 that works for sets.
This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3719): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4797). See also the remark in ru 3719 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.) |
Ref | Expression |
---|---|
ncanth | ⊢ I :V–onto→𝒫 V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ovi 6628 | . . 3 ⊢ I :V–1-1-onto→V | |
2 | f1ofo 6597 | . . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ I :V–onto→V |
4 | pwv 4797 | . . 3 ⊢ 𝒫 V = V | |
5 | foeq3 6563 | . . 3 ⊢ (𝒫 V = V → ( I :V–onto→𝒫 V ↔ I :V–onto→V)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ ( I :V–onto→𝒫 V ↔ I :V–onto→V) |
7 | 3, 6 | mpbir 234 | 1 ⊢ I :V–onto→𝒫 V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 Vcvv 3441 𝒫 cpw 4497 I cid 5424 –onto→wfo 6322 –1-1-onto→wf1o 6323 |
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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 |
This theorem is referenced by: (None) |
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