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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 12256 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
2 | eleq1 2813 | . . . . . . . . 9 ⊢ (𝑧 = 1 → (𝑧 ∈ ℕ ↔ 1 ∈ ℕ)) | |
3 | 1, 2 | mpbiri 257 | . . . . . . . 8 ⊢ (𝑧 = 1 → 𝑧 ∈ ℕ) |
4 | nnnn0 12512 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℕ0) | |
5 | dvds1 16299 | . . . . . . . . . 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 820 | . . . . . . 7 ⊢ (𝑧 = 1 ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
9 | velsn 4646 | . . . . . . 7 ⊢ (𝑧 ∈ {1} ↔ 𝑧 = 1) | |
10 | breq1 5152 | . . . . . . . 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 303 | . . . . . 6 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ 𝑧 ∈ {1}) |
13 | 12 | eqriv 2722 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} = {1} |
14 | 1ex 11242 | . . . . . 6 ⊢ 1 ∈ V | |
15 | 14 | ensn1 9042 | . . . . 5 ⊢ {1} ≈ 1o |
16 | 13, 15 | eqbrtri 5170 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 1o |
17 | 1sdom2 9265 | . . . 4 ⊢ 1o ≺ 2o | |
18 | ensdomtr 9138 | . . . 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 9002 | . . 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 16647 | . . 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 322 | 1 ⊢ ¬ 1 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3418 {csn 4630 class class class wbr 5149 1oc1o 8480 2oc2o 8481 ≈ cen 8961 ≺ csdm 8963 1c1 11141 ℕcn 12245 ℕ0cn0 12505 ∥ cdvds 16234 ℙcprime 16645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-prm 16646 |
This theorem is referenced by: isprm2 16656 nprmdvds1 16680 prm23lt5 16786 pcmpt 16864 prmo1 17009 prmlem1a 17079 prmcyg 19861 prmgrpsimpgd 20083 prmirredlem 21415 bposlem5 27266 2lgs 27385 chtvalz 34392 prmdvdsfmtnof1lem2 47062 lighneallem3 47084 |
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