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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 5316). Specifically, the identity function maps
the
universe onto its power class. Compare canth 7362 that works for sets.
This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3777): š« 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 š« V (see pwv 4906). See also the remark in ru 3777 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 | ā¢ I :Vāontoāš« V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ovi 6873 | . . 3 ā¢ I :Vā1-1-ontoāV | |
2 | f1ofo 6841 | . . 3 ā¢ ( I :Vā1-1-ontoāV ā I :VāontoāV) | |
3 | 1, 2 | ax-mp 5 | . 2 ā¢ I :VāontoāV |
4 | pwv 4906 | . . 3 ā¢ š« V = V | |
5 | foeq3 6804 | . . 3 ā¢ (š« V = V ā ( I :Vāontoāš« V ā I :VāontoāV)) | |
6 | 4, 5 | ax-mp 5 | . 2 ā¢ ( I :Vāontoāš« V ā I :VāontoāV) |
7 | 3, 6 | mpbir 230 | 1 ā¢ I :Vāontoāš« V |
Colors of variables: wff setvar class |
Syntax hints: ā wb 205 = wceq 1542 Vcvv 3475 š« cpw 4603 I cid 5574 āontoāwfo 6542 ā1-1-ontoāwf1o 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 |
This theorem is referenced by: (None) |
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