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Mirrors > Home > MPE Home > Th. List > 1nprm | Structured version Visualization version GIF version |
Description: 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
Ref | Expression |
---|---|
1nprm | ⊢ ¬ 1 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 12171 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
2 | eleq1 2826 | . . . . . . . . 9 ⊢ (𝑧 = 1 → (𝑧 ∈ ℕ ↔ 1 ∈ ℕ)) | |
3 | 1, 2 | mpbiri 258 | . . . . . . . 8 ⊢ (𝑧 = 1 → 𝑧 ∈ ℕ) |
4 | nnnn0 12427 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℕ0) | |
5 | dvds1 16208 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ0 → (𝑧 ∥ 1 ↔ 𝑧 = 1)) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → (𝑧 ∥ 1 ↔ 𝑧 = 1)) |
7 | 6 | bicomd 222 | . . . . . . . 8 ⊢ (𝑧 ∈ ℕ → (𝑧 = 1 ↔ 𝑧 ∥ 1)) |
8 | 3, 7 | biadanii 821 | . . . . . . 7 ⊢ (𝑧 = 1 ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
9 | velsn 4607 | . . . . . . 7 ⊢ (𝑧 ∈ {1} ↔ 𝑧 = 1) | |
10 | breq1 5113 | . . . . . . . 8 ⊢ (𝑛 = 𝑧 → (𝑛 ∥ 1 ↔ 𝑧 ∥ 1)) | |
11 | 10 | elrab 3650 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
12 | 8, 9, 11 | 3bitr4ri 304 | . . . . . 6 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ 𝑧 ∈ {1}) |
13 | 12 | eqriv 2734 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} = {1} |
14 | 1ex 11158 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | ensn1 8968 | . . . . 5 ⊢ {1} ≈ 1o |
16 | 13, 15 | eqbrtri 5131 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 1o |
17 | 1sdom2 9191 | . . . 4 ⊢ 1o ≺ 2o | |
18 | ensdomtr 9064 | . . . 4 ⊢ (({𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 1o ∧ 1o ≺ 2o) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≺ 2o) | |
19 | 16, 17, 18 | mp2an 691 | . . 3 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≺ 2o |
20 | sdomnen 8928 | . . 3 ⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≺ 2o → ¬ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o) | |
21 | 19, 20 | ax-mp 5 | . 2 ⊢ ¬ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o |
22 | isprm 16556 | . . 3 ⊢ (1 ∈ ℙ ↔ (1 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o)) | |
23 | 1, 22 | mpbiran 708 | . 2 ⊢ (1 ∈ ℙ ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o) |
24 | 21, 23 | mtbir 323 | 1 ⊢ ¬ 1 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3410 {csn 4591 class class class wbr 5110 1oc1o 8410 2oc2o 8411 ≈ cen 8887 ≺ csdm 8889 1c1 11059 ℕcn 12160 ℕ0cn0 12420 ∥ cdvds 16143 ℙcprime 16554 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-prm 16555 |
This theorem is referenced by: isprm2 16565 nprmdvds1 16589 prm23lt5 16693 pcmpt 16771 prmo1 16916 prmlem1a 16986 prmcyg 19678 prmgrpsimpgd 19900 prmirredlem 20909 bposlem5 26652 2lgs 26771 chtvalz 33282 prmdvdsfmtnof1lem2 45851 lighneallem3 45873 |
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