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Theorem canth 6254
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 7690. Note that must be a set: this theorem does not hold when is too large to be a set; see ncanth 6255 for a counterexample. (Use nex 1627 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 3584 . . . 4
2 canth.1 . . . . 5
32elpw2 4616 . . . 4
41, 3mpbir 209 . . 3
5 forn 5803 . . 3
64, 5syl5eleqr 2552 . 2
7 id 22 . . . . . . . . . 10
8 fveq2 5871 . . . . . . . . . 10
97, 8eleq12d 2539 . . . . . . . . 9
109notbid 294 . . . . . . . 8
1110elrab 3257 . . . . . . 7
1211baibr 904 . . . . . 6
13 nbbn 358 . . . . . 6
1412, 13sylib 196 . . . . 5
15 eleq2 2530 . . . . 5
1614, 15nsyl 121 . . . 4
1716nrex 2912 . . 3
18 fofn 5802 . . . 4
19 fvelrnb 5920 . . . 4
2018, 19syl 16 . . 3
2117, 20mtbiri 303 . 2
226, 21pm2.65i 173 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  =wceq 1395  e.wcel 1818  E.wrex 2808  {crab 2811   cvv 3109  C_wss 3475  ~Pcpw 4012  rancrn 5005  Fnwfn 5588  -onto->wfo 5591  `cfv 5593
This theorem is referenced by:  canth2  7690  canthwdom  8026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601
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