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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnoinf | Structured version Visualization version GIF version |
Description: The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.) |
Ref | Expression |
---|---|
fmtnoinf | ⊢ ran FermatNo ∉ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmtnof1 43574 | . . . 4 ⊢ FermatNo:ℕ0–1-1→ℕ | |
2 | f1f 6568 | . . . 4 ⊢ (FermatNo:ℕ0–1-1→ℕ → FermatNo:ℕ0⟶ℕ) | |
3 | fdm 6515 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → dom FermatNo = ℕ0) | |
4 | nnssnn0 11888 | . . . . . . . 8 ⊢ ℕ ⊆ ℕ0 | |
5 | nnnfi 13322 | . . . . . . . 8 ⊢ ¬ ℕ ∈ Fin | |
6 | ssfi 8726 | . . . . . . . . . 10 ⊢ ((ℕ0 ∈ Fin ∧ ℕ ⊆ ℕ0) → ℕ ∈ Fin) | |
7 | 6 | expcom 414 | . . . . . . . . 9 ⊢ (ℕ ⊆ ℕ0 → (ℕ0 ∈ Fin → ℕ ∈ Fin)) |
8 | 7 | con3d 155 | . . . . . . . 8 ⊢ (ℕ ⊆ ℕ0 → (¬ ℕ ∈ Fin → ¬ ℕ0 ∈ Fin)) |
9 | 4, 5, 8 | mp2 9 | . . . . . . 7 ⊢ ¬ ℕ0 ∈ Fin |
10 | eleq1 2897 | . . . . . . 7 ⊢ (dom FermatNo = ℕ0 → (dom FermatNo ∈ Fin ↔ ℕ0 ∈ Fin)) | |
11 | 9, 10 | mtbiri 328 | . . . . . 6 ⊢ (dom FermatNo = ℕ0 → ¬ dom FermatNo ∈ Fin) |
12 | 3, 11 | syl 17 | . . . . 5 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ dom FermatNo ∈ Fin) |
13 | ffun 6510 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → Fun FermatNo) | |
14 | fundmfibi 8791 | . . . . . 6 ⊢ (Fun FermatNo → (FermatNo ∈ Fin ↔ dom FermatNo ∈ Fin)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (FermatNo:ℕ0⟶ℕ → (FermatNo ∈ Fin ↔ dom FermatNo ∈ Fin)) |
16 | 12, 15 | mtbird 326 | . . . 4 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ FermatNo ∈ Fin) |
17 | 1, 2, 16 | mp2b 10 | . . 3 ⊢ ¬ FermatNo ∈ Fin |
18 | nn0ex 11891 | . . . 4 ⊢ ℕ0 ∈ V | |
19 | f1dmvrnfibi 8796 | . . . . 5 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (FermatNo ∈ Fin ↔ ran FermatNo ∈ Fin)) | |
20 | 19 | notbid 319 | . . . 4 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin)) |
21 | 18, 1, 20 | mp2an 688 | . . 3 ⊢ (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin) |
22 | 17, 21 | mpbi 231 | . 2 ⊢ ¬ ran FermatNo ∈ Fin |
23 | 22 | nelir 3123 | 1 ⊢ ran FermatNo ∉ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∉ wnel 3120 Vcvv 3492 ⊆ wss 3933 dom cdm 5548 ran crn 5549 Fun wfun 6342 ⟶wf 6344 –1-1→wf1 6345 Fincfn 8497 ℕcn 11626 ℕ0cn0 11885 FermatNocfmtno 43566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-fmtno 43567 |
This theorem is referenced by: prminf2 43627 |
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