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Theorem canth 6563
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 8058. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 6564 for a counterexample. (Use nex 1728 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 3671 . . . 4 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴
2 canth.1 . . . . 5 𝐴 ∈ V
32elpw2 4793 . . . 4 ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴)
41, 3mpbir 221 . . 3 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴
5 forn 6077 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
64, 5syl5eleqr 2711 . 2 (𝐹:𝐴onto→𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
7 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
8 fveq2 6150 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
97, 8eleq12d 2698 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑦)))
109notbid 308 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹𝑥) ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
1110elrab 3351 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐹𝑦)))
1211baibr 944 . . . . . 6 (𝑦𝐴 → (¬ 𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
13 nbbn 373 . . . . . 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 2693 . . . . 5 ((𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} → (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1614, 15nsyl 135 . . . 4 (𝑦𝐴 → ¬ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)})
1716nrex 2999 . . 3 ¬ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}
18 fofn 6076 . . . 4 (𝐹:𝐴onto→𝒫 𝐴𝐹 Fn 𝐴)
19 fvelrnb 6201 . . . 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 317 . 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 1480  wcel 1992  wrex 2913  {crab 2916  Vcvv 3191  wss 3560  𝒫 cpw 4135  ran crn 5080   Fn wfn 5845  ontowfo 5848  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fo 5856  df-fv 5858
This theorem is referenced by:  canth2  8058  canthwdom  8429
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