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Mirrors > Home > ILE Home > Th. List > 1nprm | 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 | 1nen2 6755 | . 2 ⊢ ¬ 1o ≈ 2o | |
2 | 1nn 8734 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
3 | eleq1 2202 | . . . . . . . . 9 ⊢ (𝑧 = 1 → (𝑧 ∈ ℕ ↔ 1 ∈ ℕ)) | |
4 | 2, 3 | mpbiri 167 | . . . . . . . 8 ⊢ (𝑧 = 1 → 𝑧 ∈ ℕ) |
5 | nnnn0 8987 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℕ0) | |
6 | dvds1 11554 | . . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ0 → (𝑧 ∥ 1 ↔ 𝑧 = 1)) | |
7 | 5, 6 | syl 14 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → (𝑧 ∥ 1 ↔ 𝑧 = 1)) |
8 | 7 | bicomd 140 | . . . . . . . 8 ⊢ (𝑧 ∈ ℕ → (𝑧 = 1 ↔ 𝑧 ∥ 1)) |
9 | 4, 8 | biadan2 451 | . . . . . . 7 ⊢ (𝑧 = 1 ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
10 | velsn 3544 | . . . . . . 7 ⊢ (𝑧 ∈ {1} ↔ 𝑧 = 1) | |
11 | breq1 3932 | . . . . . . . 8 ⊢ (𝑛 = 𝑧 → (𝑛 ∥ 1 ↔ 𝑧 ∥ 1)) | |
12 | 11 | elrab 2840 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ (𝑧 ∈ ℕ ∧ 𝑧 ∥ 1)) |
13 | 9, 10, 12 | 3bitr4ri 212 | . . . . . 6 ⊢ (𝑧 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ↔ 𝑧 ∈ {1}) |
14 | 13 | eqriv 2136 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} = {1} |
15 | 1ex 7764 | . . . . . 6 ⊢ 1 ∈ V | |
16 | 15 | ensn1 6690 | . . . . 5 ⊢ {1} ≈ 1o |
17 | 14, 16 | eqbrtri 3949 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 1o |
18 | 17 | ensymi 6676 | . . 3 ⊢ 1o ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} |
19 | isprm 11793 | . . . 4 ⊢ (1 ∈ ℙ ↔ (1 ∈ ℕ ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o)) | |
20 | 19 | simprbi 273 | . . 3 ⊢ (1 ∈ ℙ → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o) |
21 | entr 6678 | . . 3 ⊢ ((1o ≈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 1} ≈ 2o) → 1o ≈ 2o) | |
22 | 18, 20, 21 | sylancr 410 | . 2 ⊢ (1 ∈ ℙ → 1o ≈ 2o) |
23 | 1, 22 | mto 651 | 1 ⊢ ¬ 1 ∈ ℙ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {crab 2420 {csn 3527 class class class wbr 3929 1oc1o 6306 2oc2o 6307 ≈ cen 6632 1c1 7624 ℕcn 8723 ℕ0cn0 8980 ∥ cdvds 11496 ℙcprime 11791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-1o 6313 df-2o 6314 df-er 6429 df-en 6635 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-seqfrec 10222 df-exp 10296 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-dvds 11497 df-prm 11792 |
This theorem is referenced by: isprm2 11801 nprmdvds1 11823 |
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