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Mirrors > Home > ILE Home > Th. List > oddprmge3 | GIF version |
Description: An odd prime is greater than or equal to 3. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 20-Aug-2021.) |
Ref | Expression |
---|---|
oddprmge3 | ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 3257 | . . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ) | |
2 | oddprmgt2 12128 | . . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃) | |
3 | 3z 9280 | . . . . 5 ⊢ 3 ∈ ℤ | |
4 | 3 | a1i 9 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 2 < 𝑃) → 3 ∈ ℤ) |
5 | prmz 12105 | . . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ) | |
6 | 5 | adantr 276 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 2 < 𝑃) → 𝑃 ∈ ℤ) |
7 | df-3 8977 | . . . . 5 ⊢ 3 = (2 + 1) | |
8 | 2z 9279 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
9 | zltp1le 9305 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (2 < 𝑃 ↔ (2 + 1) ≤ 𝑃)) | |
10 | 8, 5, 9 | sylancr 414 | . . . . . 6 ⊢ (𝑃 ∈ ℙ → (2 < 𝑃 ↔ (2 + 1) ≤ 𝑃)) |
11 | 10 | biimpa 296 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 2 < 𝑃) → (2 + 1) ≤ 𝑃) |
12 | 7, 11 | eqbrtrid 4038 | . . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 2 < 𝑃) → 3 ≤ 𝑃) |
13 | 4, 6, 12 | 3jca 1177 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 2 < 𝑃) → (3 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 3 ≤ 𝑃)) |
14 | 1, 2, 13 | syl2anc 411 | . 2 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (3 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 3 ≤ 𝑃)) |
15 | eluz2 9532 | . 2 ⊢ (𝑃 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 3 ≤ 𝑃)) | |
16 | 14, 15 | sylibr 134 | 1 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘3)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 ∈ wcel 2148 ∖ cdif 3126 {csn 3592 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 1c1 7811 + caddc 7813 < clt 7990 ≤ cle 7991 2c2 8968 3c3 8969 ℤcz 9251 ℤ≥cuz 9526 ℙcprime 12101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-1o 6416 df-2o 6417 df-er 6534 df-en 6740 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-seqfrec 10443 df-exp 10517 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-dvds 11790 df-prm 12102 |
This theorem is referenced by: (None) |
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