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Theorem canth2 9125
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 7356. 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 5376 . . 3 𝒫 𝐴 ∈ V
3 snelpwi 5441 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 vex 3479 . . . . . . 7 𝑥 ∈ V
54sneqr 4839 . . . . . 6 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6 sneq 4636 . . . . . 6 (𝑥 = 𝑦 → {𝑥} = {𝑦})
75, 6impbii 208 . . . . 5 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
87a1i 11 . . . 4 ((𝑥𝐴𝑦𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
93, 8dom3 8987 . . 3 ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
101, 2, 9mp2an 691 . 2 𝐴 ≼ 𝒫 𝐴
111canth 7356 . . . . 5 ¬ 𝑓:𝐴onto→𝒫 𝐴
12 f1ofo 6836 . . . . 5 (𝑓:𝐴1-1-onto→𝒫 𝐴𝑓:𝐴onto→𝒫 𝐴)
1311, 12mto 196 . . . 4 ¬ 𝑓:𝐴1-1-onto→𝒫 𝐴
1413nex 1803 . . 3 ¬ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴
15 bren 8944 . . 3 (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴)
1614, 15mtbir 323 . 2 ¬ 𝐴 ≈ 𝒫 𝐴
17 brsdom 8966 . 2 (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
1810, 16, 17mpbir2an 710 1 𝐴 ≺ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  𝒫 cpw 4600  {csn 4626   class class class wbr 5146  ontowfo 6537  1-1-ontowf1o 6538  cen 8931  cdom 8932  csdm 8933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-en 8935  df-dom 8936  df-sdom 8937
This theorem is referenced by:  canth2g  9126  r1sdom  9764  alephsucpw2  10101  dfac13  10132  pwsdompw  10194  numthcor  10484  alephexp1  10569  pwcfsdom  10573  cfpwsdom  10574  gchac  10671  inawinalem  10679  tskcard  10771  gruina  10808  grothac  10820  rpnnen  16165  rexpen  16166  rucALT  16168  rectbntr0  24329
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