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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 5321). Specifically, the identity function maps
the
universe onto its power class. Compare canth 7385 that works for sets.
This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3789): 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 4909). See also the remark in ru 3789 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 6888 | . . 3 ⊢ I :V–1-1-onto→V | |
2 | f1ofo 6856 | . . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ I :V–onto→V |
4 | pwv 4909 | . . 3 ⊢ 𝒫 V = V | |
5 | foeq3 6819 | . . 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 231 | 1 ⊢ I :V–onto→𝒫 V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 Vcvv 3478 𝒫 cpw 4605 I cid 5582 –onto→wfo 6561 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: (None) |
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