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Mirrors > Home > ILE Home > Th. List > prm2orodd | GIF version |
Description: A prime number is either 2 or odd. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
prm2orodd | ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 8994 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | dvdsprime 11999 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 2 ∈ ℕ) → (2 ∥ 𝑃 ↔ (2 = 𝑃 ∨ 2 = 1))) | |
3 | 1, 2 | mpan2 422 | . . . 4 ⊢ (𝑃 ∈ ℙ → (2 ∥ 𝑃 ↔ (2 = 𝑃 ∨ 2 = 1))) |
4 | eqcom 2159 | . . . . . 6 ⊢ (2 = 𝑃 ↔ 𝑃 = 2) | |
5 | 4 | biimpi 119 | . . . . 5 ⊢ (2 = 𝑃 → 𝑃 = 2) |
6 | 1ne2 9039 | . . . . . 6 ⊢ 1 ≠ 2 | |
7 | necom 2411 | . . . . . . 7 ⊢ (1 ≠ 2 ↔ 2 ≠ 1) | |
8 | eqneqall 2337 | . . . . . . . 8 ⊢ (2 = 1 → (2 ≠ 1 → 𝑃 = 2)) | |
9 | 8 | com12 30 | . . . . . . 7 ⊢ (2 ≠ 1 → (2 = 1 → 𝑃 = 2)) |
10 | 7, 9 | sylbi 120 | . . . . . 6 ⊢ (1 ≠ 2 → (2 = 1 → 𝑃 = 2)) |
11 | 6, 10 | ax-mp 5 | . . . . 5 ⊢ (2 = 1 → 𝑃 = 2) |
12 | 5, 11 | jaoi 706 | . . . 4 ⊢ ((2 = 𝑃 ∨ 2 = 1) → 𝑃 = 2) |
13 | 3, 12 | syl6bi 162 | . . 3 ⊢ (𝑃 ∈ ℙ → (2 ∥ 𝑃 → 𝑃 = 2)) |
14 | 13 | con3d 621 | . 2 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → ¬ 2 ∥ 𝑃)) |
15 | prmz 11988 | . . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
16 | 2z 9195 | . . . 4 ⊢ 2 ∈ ℤ | |
17 | zdceq 9239 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑃 = 2) | |
18 | 16, 17 | mpan2 422 | . . 3 ⊢ (𝑃 ∈ ℤ → DECID 𝑃 = 2) |
19 | dfordc 878 | . . 3 ⊢ (DECID 𝑃 = 2 → ((𝑃 = 2 ∨ ¬ 2 ∥ 𝑃) ↔ (¬ 𝑃 = 2 → ¬ 2 ∥ 𝑃))) | |
20 | 15, 18, 19 | 3syl 17 | . 2 ⊢ (𝑃 ∈ ℙ → ((𝑃 = 2 ∨ ¬ 2 ∥ 𝑃) ↔ (¬ 𝑃 = 2 → ¬ 2 ∥ 𝑃))) |
21 | 14, 20 | mpbird 166 | 1 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∨ wo 698 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 class class class wbr 3965 1c1 7733 ℕcn 8833 2c2 8884 ℤcz 9167 ∥ cdvds 11683 ℙcprime 11984 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-1o 6363 df-2o 6364 df-er 6480 df-en 6686 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-seqfrec 10345 df-exp 10419 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-dvds 11684 df-prm 11985 |
This theorem is referenced by: (None) |
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