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Theorem canth 6752
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 8270. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 6753 for a counterexample. (Use nex 1879 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 3837 . . . 4 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴
2 canth.1 . . . . 5 𝐴 ∈ V
32elpw2 4960 . . . 4 ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴)
41, 3mpbir 221 . . 3 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴
5 forn 6260 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
64, 5syl5eleqr 2857 . 2 (𝐹:𝐴onto→𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
7 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
8 fveq2 6333 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
97, 8eleq12d 2844 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑦)))
109notbid 307 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹𝑥) ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
1110elrab 3516 . . . . . . 7 (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝐹𝑦)))
1211baibr 520 . . . . . 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 2839 . . . . 5 ((𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} → (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1614, 15nsyl 137 . . . 4 (𝑦𝐴 → ¬ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)})
1716nrex 3148 . . 3 ¬ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}
18 fofn 6259 . . . 4 (𝐹:𝐴onto→𝒫 𝐴𝐹 Fn 𝐴)
19 fvelrnb 6386 . . . 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 1631  wcel 2145  wrex 3062  {crab 3065  Vcvv 3351  wss 3724  𝒫 cpw 4298  ran crn 5251   Fn wfn 6027  ontowfo 6030  cfv 6032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-fo 6038  df-fv 6040
This theorem is referenced by:  canth2  8270  canthwdom  8641
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