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Theorem canth2 8098
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 6593. 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 4839 . . 3 𝒫 𝐴 ∈ V
3 snelpwi 4903 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
4 vex 3198 . . . . . . 7 𝑥 ∈ V
54sneqr 4362 . . . . . 6 ({𝑥} = {𝑦} → 𝑥 = 𝑦)
6 sneq 4178 . . . . . 6 (𝑥 = 𝑦 → {𝑥} = {𝑦})
75, 6impbii 199 . . . . 5 ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)
87a1i 11 . . . 4 ((𝑥𝐴𝑦𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦))
93, 8dom3 7984 . . 3 ((𝐴 ∈ V ∧ 𝒫 𝐴 ∈ V) → 𝐴 ≼ 𝒫 𝐴)
101, 2, 9mp2an 707 . 2 𝐴 ≼ 𝒫 𝐴
111canth 6593 . . . . 5 ¬ 𝑓:𝐴onto→𝒫 𝐴
12 f1ofo 6131 . . . . 5 (𝑓:𝐴1-1-onto→𝒫 𝐴𝑓:𝐴onto→𝒫 𝐴)
1311, 12mto 188 . . . 4 ¬ 𝑓:𝐴1-1-onto→𝒫 𝐴
1413nex 1729 . . 3 ¬ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴
15 bren 7949 . . 3 (𝐴 ≈ 𝒫 𝐴 ↔ ∃𝑓 𝑓:𝐴1-1-onto→𝒫 𝐴)
1614, 15mtbir 313 . 2 ¬ 𝐴 ≈ 𝒫 𝐴
17 brsdom 7963 . 2 (𝐴 ≺ 𝒫 𝐴 ↔ (𝐴 ≼ 𝒫 𝐴 ∧ ¬ 𝐴 ≈ 𝒫 𝐴))
1810, 16, 17mpbir2an 954 1 𝐴 ≺ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  𝒫 cpw 4149  {csn 4168   class class class wbr 4644  ontowfo 5874  1-1-ontowf1o 5875  cen 7937  cdom 7938  csdm 7939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-en 7941  df-dom 7942  df-sdom 7943
This theorem is referenced by:  canth2g  8099  r1sdom  8622  alephsucpw2  8919  dfac13  8949  pwsdompw  9011  numthcor  9301  alephexp1  9386  pwcfsdom  9390  cfpwsdom  9391  gchhar  9486  gchac  9488  inawinalem  9496  tskcard  9588  gruina  9625  grothac  9637  rpnnen  14937  rexpen  14938  rucALT  14940  rectbntr0  22616
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