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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 43099 | . . . 4 ⊢ FermatNo:ℕ0–1-1→ℕ | |
2 | f1f 6401 | . . . 4 ⊢ (FermatNo:ℕ0–1-1→ℕ → FermatNo:ℕ0⟶ℕ) | |
3 | fdm 6349 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → dom FermatNo = ℕ0) | |
4 | nnssnn0 11708 | . . . . . . . 8 ⊢ ℕ ⊆ ℕ0 | |
5 | nnnfi 13147 | . . . . . . . 8 ⊢ ¬ ℕ ∈ Fin | |
6 | ssfi 8531 | . . . . . . . . . 10 ⊢ ((ℕ0 ∈ Fin ∧ ℕ ⊆ ℕ0) → ℕ ∈ Fin) | |
7 | 6 | expcom 406 | . . . . . . . . 9 ⊢ (ℕ ⊆ ℕ0 → (ℕ0 ∈ Fin → ℕ ∈ Fin)) |
8 | 7 | con3d 150 | . . . . . . . 8 ⊢ (ℕ ⊆ ℕ0 → (¬ ℕ ∈ Fin → ¬ ℕ0 ∈ Fin)) |
9 | 4, 5, 8 | mp2 9 | . . . . . . 7 ⊢ ¬ ℕ0 ∈ Fin |
10 | eleq1 2846 | . . . . . . 7 ⊢ (dom FermatNo = ℕ0 → (dom FermatNo ∈ Fin ↔ ℕ0 ∈ Fin)) | |
11 | 9, 10 | mtbiri 319 | . . . . . 6 ⊢ (dom FermatNo = ℕ0 → ¬ dom FermatNo ∈ Fin) |
12 | 3, 11 | syl 17 | . . . . 5 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ dom FermatNo ∈ Fin) |
13 | ffun 6344 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → Fun FermatNo) | |
14 | fundmfibi 8596 | . . . . . 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 317 | . . . 4 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ FermatNo ∈ Fin) |
17 | 1, 2, 16 | mp2b 10 | . . 3 ⊢ ¬ FermatNo ∈ Fin |
18 | nn0ex 11712 | . . . 4 ⊢ ℕ0 ∈ V | |
19 | f1dmvrnfibi 8601 | . . . . 5 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (FermatNo ∈ Fin ↔ ran FermatNo ∈ Fin)) | |
20 | 19 | notbid 310 | . . . 4 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin)) |
21 | 18, 1, 20 | mp2an 680 | . . 3 ⊢ (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin) |
22 | 17, 21 | mpbi 222 | . 2 ⊢ ¬ ran FermatNo ∈ Fin |
23 | 22 | nelir 3069 | 1 ⊢ ran FermatNo ∉ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∉ wnel 3066 Vcvv 3408 ⊆ wss 3822 dom cdm 5403 ran crn 5404 Fun wfun 6179 ⟶wf 6181 –1-1→wf1 6182 Fincfn 8304 ℕcn 11437 ℕ0cn0 11705 FermatNocfmtno 43091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-n0 11706 df-z 11792 df-uz 12057 df-rp 12203 df-seq 13183 df-exp 13243 df-fmtno 43092 |
This theorem is referenced by: prminf2 43152 |
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