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Theorem canth 6178
 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 6899. Note that must be a set: this theorem does not hold when is too large to be a set; see ncanth 6179 for a counterexample. (Use nex 1587 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
StepHypRef Expression
1 ssrab2 3179 . . . 4
2 canth.1 . . . . 5
32elpw2 4064 . . . 4
41, 3mpbir 202 . . 3
5 forn 5311 . . 3
64, 5syl5eleqr 2340 . 2
7 id 21 . . . . . . . . . 10
8 fveq2 5377 . . . . . . . . . 10
97, 8eleq12d 2321 . . . . . . . . 9
109notbid 287 . . . . . . . 8
1110elrab 2860 . . . . . . 7
1211baibr 877 . . . . . 6
13 nbbn 349 . . . . . 6
1412, 13sylib 190 . . . . 5
15 eleq2 2314 . . . . 5
1614, 15nsyl 115 . . . 4
1716nrex 2607 . . 3
18 fofn 5310 . . . 4
19 fvelrnb 5422 . . . 4
2018, 19syl 17 . . 3
2117, 20mtbiri 296 . 2
226, 21pm2.65i 167 1
 Colors of variables: wff set class Syntax hints:   wn 5   wb 178   wceq 1619   wcel 1621  wrex 2510  crab 2512  cvv 2727   wss 3078  cpw 3530   crn 4581   wfn 4587  wfo 4590  cfv 4592 This theorem is referenced by:  canth2  6899  canthwdom  7177 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fo 4606  df-fv 4608
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