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Theorem canth 5831
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. (Use nex 1500 if you want the form ¬ ∃𝑓𝑓:𝐴onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
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 3242 . . . 4 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ⊆ 𝐴
31, 2elpwi2 4160 . . 3 {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ 𝒫 𝐴
4 forn 5443 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴)
53, 4eleqtrrid 2267 . 2 (𝐹:𝐴onto→𝒫 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
6 pm5.19 706 . . . . . 6 ¬ (𝑦 ∈ (𝐹𝑦) ↔ ¬ 𝑦 ∈ (𝐹𝑦))
7 eleq2 2241 . . . . . . 7 ((𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} → (𝑦 ∈ (𝐹𝑦) ↔ 𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
8 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
9 fveq2 5517 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
108, 9eleq12d 2248 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹𝑦)))
1110notbid 667 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹𝑥) ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
1211elrab3 2896 . . . . . . 7 (𝑦𝐴 → (𝑦 ∈ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
137, 12sylan9bbr 463 . . . . . 6 ((𝑦𝐴 ∧ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}) → (𝑦 ∈ (𝐹𝑦) ↔ ¬ 𝑦 ∈ (𝐹𝑦)))
146, 13mto 662 . . . . 5 ¬ (𝑦𝐴 ∧ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)})
1514imnani 691 . . . 4 (𝑦𝐴 → ¬ (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)})
1615nrex 2569 . . 3 ¬ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}
17 fofn 5442 . . . 4 (𝐹:𝐴onto→𝒫 𝐴𝐹 Fn 𝐴)
18 fvelrnb 5565 . . . 4 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
1917, 18syl 14 . . 3 (𝐹:𝐴onto→𝒫 𝐴 → ({𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)}))
2016, 19mtbiri 675 . 2 (𝐹:𝐴onto→𝒫 𝐴 → ¬ {𝑥𝐴 ∣ ¬ 𝑥 ∈ (𝐹𝑥)} ∈ ran 𝐹)
215, 20pm2.65i 639 1 ¬ 𝐹:𝐴onto→𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105   = wceq 1353  wcel 2148  wrex 2456  {crab 2459  Vcvv 2739  𝒫 cpw 3577  ran crn 4629   Fn wfn 5213  ontowfo 5216  cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226
This theorem is referenced by: (None)
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