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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 39985 | . . . 4 ⊢ FermatNo:ℕ0–1-1→ℕ | |
2 | f1f 6014 | . . . 4 ⊢ (FermatNo:ℕ0–1-1→ℕ → FermatNo:ℕ0⟶ℕ) | |
3 | fdm 5964 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → dom FermatNo = ℕ0) | |
4 | nnssnn0 11172 | . . . . . . . 8 ⊢ ℕ ⊆ ℕ0 | |
5 | nnnfi 12627 | . . . . . . . 8 ⊢ ¬ ℕ ∈ Fin | |
6 | ssfi 8065 | . . . . . . . . . 10 ⊢ ((ℕ0 ∈ Fin ∧ ℕ ⊆ ℕ0) → ℕ ∈ Fin) | |
7 | 6 | expcom 450 | . . . . . . . . 9 ⊢ (ℕ ⊆ ℕ0 → (ℕ0 ∈ Fin → ℕ ∈ Fin)) |
8 | 7 | con3d 147 | . . . . . . . 8 ⊢ (ℕ ⊆ ℕ0 → (¬ ℕ ∈ Fin → ¬ ℕ0 ∈ Fin)) |
9 | 4, 5, 8 | mp2 9 | . . . . . . 7 ⊢ ¬ ℕ0 ∈ Fin |
10 | eleq1 2676 | . . . . . . 7 ⊢ (dom FermatNo = ℕ0 → (dom FermatNo ∈ Fin ↔ ℕ0 ∈ Fin)) | |
11 | 9, 10 | mtbiri 316 | . . . . . 6 ⊢ (dom FermatNo = ℕ0 → ¬ dom FermatNo ∈ Fin) |
12 | 3, 11 | syl 17 | . . . . 5 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ dom FermatNo ∈ Fin) |
13 | ffun 5961 | . . . . . 6 ⊢ (FermatNo:ℕ0⟶ℕ → Fun FermatNo) | |
14 | fundmfibi 8130 | . . . . . 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 314 | . . . 4 ⊢ (FermatNo:ℕ0⟶ℕ → ¬ FermatNo ∈ Fin) |
17 | 1, 2, 16 | mp2b 10 | . . 3 ⊢ ¬ FermatNo ∈ Fin |
18 | nn0ex 11175 | . . . 4 ⊢ ℕ0 ∈ V | |
19 | f1dmvrnfibi 8133 | . . . . 5 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (FermatNo ∈ Fin ↔ ran FermatNo ∈ Fin)) | |
20 | 19 | notbid 307 | . . . 4 ⊢ ((ℕ0 ∈ V ∧ FermatNo:ℕ0–1-1→ℕ) → (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin)) |
21 | 18, 1, 20 | mp2an 704 | . . 3 ⊢ (¬ FermatNo ∈ Fin ↔ ¬ ran FermatNo ∈ Fin) |
22 | 17, 21 | mpbi 219 | . 2 ⊢ ¬ ran FermatNo ∈ Fin |
23 | 22 | nelir 2886 | 1 ⊢ ran FermatNo ∉ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 Vcvv 3173 ⊆ wss 3540 dom cdm 5038 ran crn 5039 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 Fincfn 7841 ℕcn 10897 ℕ0cn0 11169 FermatNocfmtno 39977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-fmtno 39978 |
This theorem is referenced by: prminf2 40038 |
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