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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 45817 | . . . 4 ⊢ FermatNo:ℕ0–1-1→ℕ | |
2 | f1f 6742 | . . . 4 ⊢ (FermatNo:ℕ0–1-1→ℕ → FermatNo:ℕ0⟶ℕ) | |
3 | fdm 6681 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → dom FermatNo = ℕ0) | |
4 | nnssnn0 12424 | . . . . . . . 8 ⊢ ℕ ⊆ ℕ0 | |
5 | nnnfi 13880 | . . . . . . . 8 ⊢ ¬ ℕ ∈ Fin | |
6 | ssfi 9123 | . . . . . . . . . 10 ⊢ ((ℕ0 ∈ Fin ∧ ℕ ⊆ ℕ0) → ℕ ∈ Fin) | |
7 | 6 | expcom 415 | . . . . . . . . 9 ⊢ (ℕ ⊆ ℕ0 → (ℕ0 ∈ Fin → ℕ ∈ Fin)) |
8 | 7 | con3d 152 | . . . . . . . 8 ⊢ (ℕ ⊆ ℕ0 → (¬ ℕ ∈ Fin → ¬ ℕ0 ∈ Fin)) |
9 | 4, 5, 8 | mp2 9 | . . . . . . 7 ⊢ ¬ ℕ0 ∈ Fin |
10 | eleq1 2822 | . . . . . . 7 ⊢ (dom FermatNo = ℕ0 → (dom FermatNo ∈ Fin ↔ ℕ0 ∈ Fin)) | |
11 | 9, 10 | mtbiri 327 | . . . . . 6 ⊢ (dom FermatNo = ℕ0 → ¬ dom FermatNo ∈ Fin) |
12 | 3, 11 | syl 17 | . . . . 5 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ dom FermatNo ∈ Fin) |
13 | ffun 6675 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → Fun FermatNo) | |
14 | fundmfibi 9281 | . . . . . 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 325 | . . . 4 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ FermatNo ∈ Fin) |
17 | 1, 2, 16 | mp2b 10 | . . 3 ⊢ ¬ FermatNo ∈ Fin |
18 | nn0ex 12427 | . . . 4 ⊢ ℕ0 ∈ V | |
19 | f1dmvrnfibi 9286 | . . . . 5 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (FermatNo ∈ Fin ↔ ran FermatNo ∈ Fin)) | |
20 | 19 | notbid 318 | . . . 4 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin)) |
21 | 18, 1, 20 | mp2an 691 | . . 3 ⊢ (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin) |
22 | 17, 21 | mpbi 229 | . 2 ⊢ ¬ ran FermatNo ∈ Fin |
23 | 22 | nelir 3049 | 1 ⊢ ran FermatNo ∉ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∉ wnel 3046 Vcvv 3447 ⊆ wss 3914 dom cdm 5637 ran crn 5638 Fun wfun 6494 ⟶wf 6496 –1-1→wf1 6497 Fincfn 8889 ℕcn 12161 ℕ0cn0 12421 FermatNocfmtno 45809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-seq 13916 df-exp 13977 df-fmtno 45810 |
This theorem is referenced by: prminf2 45870 |
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