MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canth Structured version   Visualization version   GIF version

Theorem canth 6648
Description: No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e. no function can map 𝐴 it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 8154. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 6649 for a counterexample. (Use nex 1771 if you want the form ¬ ∃𝑓𝑓:𝐴onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
Hypothesis
Ref Expression
canth.1 𝐴 ∈ V
Assertion
Ref Expression
canth ¬ 𝐹:𝐴onto→𝒫 𝐴

Proof of Theorem canth
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3720 . . . 4 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴
2 canth.1 . . . . 5 𝐴 ∈ V
32elpw2 4858 . . . 4 ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴)
41, 3mpbir 221 . . 3 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴
5 forn 6156 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
64, 5syl5eleqr 2737 . 2 (𝐹:𝐴onto→𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
7 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
8 fveq2 6229 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
97, 8eleq12d 2724 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑦)))
109notbid 307 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹𝑥) ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
1110elrab 3396 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐹𝑦)))
1211baibr 965 . . . . . 6 (𝑦𝐴 → (¬ 𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
13 nbbn 372 . . . . . 6 ((¬ 𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}) ↔ ¬ (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1412, 13sylib 208 . . . . 5 (𝑦𝐴 → ¬ (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
15 eleq2 2719 . . . . 5 ((𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} → (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1614, 15nsyl 135 . . . 4 (𝑦𝐴 → ¬ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)})
1716nrex 3029 . . 3 ¬ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}
18 fofn 6155 . . . 4 (𝐹:𝐴onto→𝒫 𝐴𝐹 Fn 𝐴)
19 fvelrnb 6282 . . . 4 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
2018, 19syl 17 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
2117, 20mtbiri 316 . 2 (𝐹:𝐴onto→𝒫 𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
226, 21pm2.65i 185 1 ¬ 𝐹:𝐴onto→𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  ran crn 5144   Fn wfn 5921  ontowfo 5924  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934
This theorem is referenced by:  canth2  8154  canthwdom  8525
  Copyright terms: Public domain W3C validator