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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 5221). Specifically, the identity function maps
the
universe onto its power class. Compare canth 7113 that works for sets.
This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3773): 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 4837). See also the remark in ru 3773 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 6655 | . . 3 ⊢ I :V–1-1-onto→V | |
2 | f1ofo 6624 | . . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ I :V–onto→V |
4 | pwv 4837 | . . 3 ⊢ 𝒫 V = V | |
5 | foeq3 6590 | . . 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 233 | 1 ⊢ I :V–onto→𝒫 V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 Vcvv 3496 𝒫 cpw 4541 I cid 5461 –onto→wfo 6355 –1-1-onto→wf1o 6356 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 |
This theorem is referenced by: (None) |
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