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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 11643 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
2 | eleq1 2900 | . . . . . . . . 9 ⊢ (𝑧 = 1 → (𝑧 ∈ ℕ ↔ 1 ∈ ℕ)) | |
3 | 1, 2 | mpbiri 260 | . . . . . . . 8 ⊢ (𝑧 = 1 → 𝑧 ∈ ℕ) |
4 | nnnn0 11898 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℕ0) | |
5 | dvds1 15663 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ0 → (𝑧 ∥ 1 ↔ 𝑧 = 1)) | |
6 | 4, 5 | syl 17 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → (𝑧 ∥ 1 ↔ 𝑧 = 1)) |
7 | 6 | bicomd 225 | . . . . . . . 8 ⊢ (𝑧 ∈ ℕ → (𝑧 = 1 ↔ 𝑧 ∥ 1)) |
8 | 3, 7 | biadanii 820 | . . . . . . 7 ⊢ (𝑧 = 1 ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
9 | velsn 4576 | . . . . . . 7 ⊢ (𝑧 ∈ {1} ↔ 𝑧 = 1) | |
10 | breq1 5061 | . . . . . . . 8 ⊢ (𝑛 = 𝑧 → (𝑛 ∥ 1 ↔ 𝑧 ∥ 1)) | |
11 | 10 | elrab 3679 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
12 | 8, 9, 11 | 3bitr4ri 306 | . . . . . 6 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ 𝑧 ∈ {1}) |
13 | 12 | eqriv 2818 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} = {1} |
14 | 1ex 10631 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | ensn1 8567 | . . . . 5 ⊢ {1} ≈ 1o |
16 | 13, 15 | eqbrtri 5079 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 1o |
17 | 1sdom2 8711 | . . . 4 ⊢ 1o ≺ 2o | |
18 | ensdomtr 8647 | . . . 4 ⊢ (({𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 1o ∧ 1o ≺ 2o) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≺ 2o) | |
19 | 16, 17, 18 | mp2an 690 | . . 3 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≺ 2o |
20 | sdomnen 8532 | . . 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 16011 | . . 3 ⊢ (1 ∈ ℙ ↔ (1 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o)) | |
23 | 1, 22 | mpbiran 707 | . 2 ⊢ (1 ∈ ℙ ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o) |
24 | 21, 23 | mtbir 325 | 1 ⊢ ¬ 1 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 {csn 4560 class class class wbr 5058 1oc1o 8089 2oc2o 8090 ≈ cen 8500 ≺ csdm 8502 1c1 10532 ℕcn 11632 ℕ0cn0 11891 ∥ cdvds 15601 ℙcprime 16009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-prm 16010 |
This theorem is referenced by: isprm2 16020 nprmdvds1 16044 prm23lt5 16145 pcmpt 16222 prmo1 16367 prmlem1a 16434 prmcyg 19008 prmgrpsimpgd 19230 prmirredlem 20634 bposlem5 25858 2lgs 25977 chtvalz 31895 prmdvdsfmtnof1lem2 43741 lighneallem3 43766 |
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