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Theorem canth 7384
Description: No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e., no function can map 𝐴 onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 9168. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 7385 for a counterexample. (Use nex 1796 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 canth.1 . . . 4 𝐴 ∈ V
2 ssrab2 4089 . . . 4 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴
31, 2elpwi2 5340 . . 3 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴
4 forn 6823 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
53, 4eleqtrrid 2845 . 2 (𝐹:𝐴onto→𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
6 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
7 fveq2 6906 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
86, 7eleq12d 2832 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑦)))
98notbid 318 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹𝑥) ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
109elrab 3694 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐹𝑦)))
1110baibr 536 . . . . . 6 (𝑦𝐴 → (¬ 𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
12 nbbn 383 . . . . . 6 ((¬ 𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}) ↔ ¬ (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1311, 12sylib 218 . . . . 5 (𝑦𝐴 → ¬ (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
14 eleq2 2827 . . . . 5 ((𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} → (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1513, 14nsyl 140 . . . 4 (𝑦𝐴 → ¬ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)})
1615nrex 3071 . . 3 ¬ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}
17 fofn 6822 . . . 4 (𝐹:𝐴onto→𝒫 𝐴𝐹 Fn 𝐴)
18 fvelrnb 6968 . . . 4 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1917, 18syl 17 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
2016, 19mtbiri 327 . 2 (𝐹:𝐴onto→𝒫 𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
215, 20pm2.65i 194 1 ¬ 𝐹:𝐴onto→𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1536  wcel 2105  wrex 3067  {crab 3432  Vcvv 3477  𝒫 cpw 4604  ran crn 5689   Fn wfn 6557  ontowfo 6560  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570
This theorem is referenced by:  canth2  9168  canthwdom  9616
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