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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 9039 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | dvdsprime 12076 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 2 ∈ ℕ) → (2 ∥ 𝑃 ↔ (2 = 𝑃 ∨ 2 = 1))) | |
3 | 1, 2 | mpan2 423 | . . . 4 ⊢ (𝑃 ∈ ℙ → (2 ∥ 𝑃 ↔ (2 = 𝑃 ∨ 2 = 1))) |
4 | eqcom 2172 | . . . . . 6 ⊢ (2 = 𝑃 ↔ 𝑃 = 2) | |
5 | 4 | biimpi 119 | . . . . 5 ⊢ (2 = 𝑃 → 𝑃 = 2) |
6 | 1ne2 9084 | . . . . . 6 ⊢ 1 ≠ 2 | |
7 | necom 2424 | . . . . . . 7 ⊢ (1 ≠ 2 ↔ 2 ≠ 1) | |
8 | eqneqall 2350 | . . . . . . . 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 711 | . . . 4 ⊢ ((2 = 𝑃 ∨ 2 = 1) → 𝑃 = 2) |
13 | 3, 12 | syl6bi 162 | . . 3 ⊢ (𝑃 ∈ ℙ → (2 ∥ 𝑃 → 𝑃 = 2)) |
14 | 13 | con3d 626 | . 2 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → ¬ 2 ∥ 𝑃)) |
15 | prmz 12065 | . . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
16 | 2z 9240 | . . . 4 ⊢ 2 ∈ ℤ | |
17 | zdceq 9287 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → DECID 𝑃 = 2) | |
18 | 16, 17 | mpan2 423 | . . 3 ⊢ (𝑃 ∈ ℤ → DECID 𝑃 = 2) |
19 | dfordc 887 | . . 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 703 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 class class class wbr 3989 1c1 7775 ℕcn 8878 2c2 8929 ℤcz 9212 ∥ cdvds 11749 ℙcprime 12061 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-prm 12062 |
This theorem is referenced by: lgsval 13699 lgsfvalg 13700 |
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