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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 5322). Specifically, the identity function maps
the
universe onto its power class. Compare canth 7379 that works for sets.
This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3774): 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 4912). See also the remark in ru 3774 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 6884 | . . 3 ⊢ I :V–1-1-onto→V | |
2 | f1ofo 6852 | . . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ I :V–onto→V |
4 | pwv 4912 | . . 3 ⊢ 𝒫 V = V | |
5 | foeq3 6815 | . . 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 230 | 1 ⊢ I :V–onto→𝒫 V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 Vcvv 3462 𝒫 cpw 4607 I cid 5581 –onto→wfo 6554 –1-1-onto→wf1o 6555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-br 5156 df-opab 5218 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 |
This theorem is referenced by: (None) |
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