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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 5034). Specifically, the identity function maps the universe onto its power class. Compare canth 6880 that works for sets. See also the remark in ru 3651 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.) |
Ref | Expression |
---|---|
ncanth | ⊢ I :V–onto→𝒫 V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ovi 6429 | . . 3 ⊢ I :V–1-1-onto→V | |
2 | pwv 4668 | . . . 4 ⊢ 𝒫 V = V | |
3 | f1oeq3 6382 | . . . 4 ⊢ (𝒫 V = V → ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ( I :V–1-1-onto→𝒫 V ↔ I :V–1-1-onto→V) |
5 | 1, 4 | mpbir 223 | . 2 ⊢ I :V–1-1-onto→𝒫 V |
6 | f1ofo 6398 | . 2 ⊢ ( I :V–1-1-onto→𝒫 V → I :V–onto→𝒫 V) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ I :V–onto→𝒫 V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1601 Vcvv 3398 𝒫 cpw 4379 I cid 5260 –onto→wfo 6133 –1-1-onto→wf1o 6134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 |
This theorem is referenced by: (None) |
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