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Theorem canth 6575
 Description: No set is equinumerous to its power set (Cantor's theorem), i.e. no function can map it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 7296. Note that must be a set: this theorem does not hold when is too large to be a set; see ncanth 6576 for a counterexample. (Use nex 1565 if you want the form .) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
Hypothesis
Ref Expression
canth.1
Assertion
Ref Expression
canth

Proof of Theorem canth
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3417 . . . 4
2 canth.1 . . . . 5
32elpw2 4399 . . . 4
41, 3mpbir 202 . . 3
5 forn 5691 . . 3
64, 5syl5eleqr 2530 . 2
7 id 21 . . . . . . . . . 10
8 fveq2 5763 . . . . . . . . . 10
97, 8eleq12d 2511 . . . . . . . . 9
109notbid 287 . . . . . . . 8
1110elrab 3101 . . . . . . 7
1211baibr 874 . . . . . 6
13 nbbn 349 . . . . . 6
1412, 13sylib 190 . . . . 5
15 eleq2 2504 . . . . 5
1614, 15nsyl 116 . . . 4
1716nrex 2815 . . 3
18 fofn 5690 . . . 4
19 fvelrnb 5810 . . . 4
2018, 19syl 16 . . 3
2117, 20mtbiri 296 . 2
226, 21pm2.65i 168 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 178   wceq 1654   wcel 1728  wrex 2713  crab 2716  cvv 2965   wss 3309  cpw 3828   crn 4914   wfn 5484  wfo 5487  cfv 5489 This theorem is referenced by:  canth2  7296  canthwdom  7583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fo 5495  df-fv 5497
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