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Mirrors > Home > ILE Home > Th. List > canth | GIF version |
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 1487 if you want the form ¬ ∃𝑓𝑓:𝐴–onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.) |
Ref | Expression |
---|---|
canth.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
canth | ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | canth.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | ssrab2 3222 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ⊆ 𝐴 | |
3 | 1, 2 | elpwi2 4131 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ 𝒫 𝐴 |
4 | forn 5407 | . . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ran 𝐹 = 𝒫 𝐴) | |
5 | 3, 4 | eleqtrrid 2254 | . 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹) |
6 | pm5.19 696 | . . . . . 6 ⊢ ¬ (𝑦 ∈ (𝐹‘𝑦) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦)) | |
7 | eleq2 2228 | . . . . . . 7 ⊢ ((𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} → (𝑦 ∈ (𝐹‘𝑦) ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})) | |
8 | id 19 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
9 | fveq2 5480 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
10 | 8, 9 | eleq12d 2235 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘𝑥) ↔ 𝑦 ∈ (𝐹‘𝑦))) |
11 | 10 | notbid 657 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝐹‘𝑥) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦))) |
12 | 11 | elrab3 2878 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ↔ ¬ 𝑦 ∈ (𝐹‘𝑦))) |
13 | 7, 12 | sylan9bbr 459 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) → (𝑦 ∈ (𝐹‘𝑦) ↔ ¬ 𝑦 ∈ (𝐹‘𝑦))) |
14 | 6, 13 | mto 652 | . . . . 5 ⊢ ¬ (𝑦 ∈ 𝐴 ∧ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) |
15 | 14 | imnani 681 | . . . 4 ⊢ (𝑦 ∈ 𝐴 → ¬ (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)}) |
16 | 15 | nrex 2556 | . . 3 ⊢ ¬ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} |
17 | fofn 5406 | . . . 4 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → 𝐹 Fn 𝐴) | |
18 | fvelrnb 5528 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})) | |
19 | 17, 18 | syl 14 | . . 3 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)})) |
20 | 16, 19 | mtbiri 665 | . 2 ⊢ (𝐹:𝐴–onto→𝒫 𝐴 → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ (𝐹‘𝑥)} ∈ ran 𝐹) |
21 | 5, 20 | pm2.65i 629 | 1 ⊢ ¬ 𝐹:𝐴–onto→𝒫 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 {crab 2446 Vcvv 2721 𝒫 cpw 3553 ran crn 4599 Fn wfn 5177 –onto→wfo 5180 ‘cfv 5182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 |
This theorem is referenced by: (None) |
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