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Theorem canth2 7998
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 6508. 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 4774 . . 3 𝒫 𝐴 ∈ V
3 snelpwi 4839 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 vex 3176 . . . . . . 7 𝑥 ∈ V
54sneqr 4311 . . . . . 6 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6 sneq 4135 . . . . . 6 (𝑥 = 𝑦 → {𝑥} = {𝑦})
75, 6impbii 198 . . . . 5 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
87a1i 11 . . . 4 ((𝑥𝐴𝑦𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
93, 8dom3 7885 . . 3 ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
101, 2, 9mp2an 704 . 2 𝐴 ≼ 𝒫 𝐴
111canth 6508 . . . . 5 ¬ 𝑓:𝐴onto→𝒫 𝐴
12 f1ofo 6057 . . . . 5 (𝑓:𝐴1-1-onto→𝒫 𝐴𝑓:𝐴onto→𝒫 𝐴)
1311, 12mto 187 . . . 4 ¬ 𝑓:𝐴1-1-onto→𝒫 𝐴
1413nex 1722 . . 3 ¬ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴
15 bren 7850 . . 3 (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴)
1614, 15mtbir 312 . 2 ¬ 𝐴 ≈ 𝒫 𝐴
17 brsdom 7864 . 2 (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
1810, 16, 17mpbir2an 957 1 𝐴 ≺ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977  Vcvv 3173  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  ontowfo 5802  1-1-ontowf1o 5803  cen 7838  cdom 7839  csdm 7840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-en 7842  df-dom 7843  df-sdom 7844
This theorem is referenced by:  canth2g  7999  r1sdom  8520  alephsucpw2  8817  dfac13  8847  pwsdompw  8909  numthcor  9199  alephexp1  9280  pwcfsdom  9284  cfpwsdom  9285  gchhar  9380  gchac  9382  inawinalem  9390  tskcard  9482  gruina  9519  grothac  9531  rpnnen  14795  rexpen  14796  rucALT  14798  rectbntr0  22443
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