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| Description: Cantor's theorem fails
for the universal class (which is not a set but a
     proper class by vprc 5314).  Specifically, the identity function maps
the
     universe onto its power class.  Compare canth 7386 that works for sets. This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3785): 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 4903). See also the remark in ru 3785 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 6886 | . . 3 ⊢ I :V–1-1-onto→V | |
| 2 | f1ofo 6854 | . . 3 ⊢ ( I :V–1-1-onto→V → I :V–onto→V) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ I :V–onto→V | 
| 4 | pwv 4903 | . . 3 ⊢ 𝒫 V = V | |
| 5 | foeq3 6817 | . . 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 1539 Vcvv 3479 𝒫 cpw 4599 I cid 5576 –onto→wfo 6558 –1-1-onto→wf1o 6559 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 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 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 | 
| This theorem is referenced by: (None) | 
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