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Theorem canth2 8269
Description: Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 6751. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
Hypothesis
Ref Expression
canth2.1 𝐴 ∈ V
Assertion
Ref Expression
canth2 𝐴 ≺ 𝒫 𝐴

Proof of Theorem canth2
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 canth2.1 . . 3 𝐴 ∈ V
21pwex 4981 . . 3 𝒫 𝐴 ∈ V
3 snelpwi 5040 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 vex 3354 . . . . . . 7 𝑥 ∈ V
54sneqr 4504 . . . . . 6 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6 sneq 4326 . . . . . 6 (𝑥 = 𝑦 → {𝑥} = {𝑦})
75, 6impbii 199 . . . . 5 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
87a1i 11 . . . 4 ((𝑥𝐴𝑦𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
93, 8dom3 8153 . . 3 ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
101, 2, 9mp2an 672 . 2 𝐴 ≼ 𝒫 𝐴
111canth 6751 . . . . 5 ¬ 𝑓:𝐴onto→𝒫 𝐴
12 f1ofo 6285 . . . . 5 (𝑓:𝐴1-1-onto→𝒫 𝐴𝑓:𝐴onto→𝒫 𝐴)
1311, 12mto 188 . . . 4 ¬ 𝑓:𝐴1-1-onto→𝒫 𝐴
1413nex 1879 . . 3 ¬ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴
15 bren 8118 . . 3 (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴)
1614, 15mtbir 312 . 2 ¬ 𝐴 ≈ 𝒫 𝐴
17 brsdom 8132 . 2 (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
1810, 16, 17mpbir2an 690 1 𝐴 ≺ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  𝒫 cpw 4297  {csn 4316   class class class wbr 4786  ontowfo 6029  1-1-ontowf1o 6030  cen 8106  cdom 8107  csdm 8108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-en 8110  df-dom 8111  df-sdom 8112
This theorem is referenced by:  canth2g  8270  r1sdom  8801  alephsucpw2  9134  dfac13  9166  pwsdompw  9228  numthcor  9518  alephexp1  9603  pwcfsdom  9607  cfpwsdom  9608  gchhar  9703  gchac  9705  inawinalem  9713  tskcard  9805  gruina  9842  grothac  9854  rpnnen  15162  rexpen  15163  rucALT  15165  rectbntr0  22855
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