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Mirrors > Home > MPE Home > Th. List > Mathboxes > evenprm2 | Structured version Visualization version GIF version |
Description: A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.) |
Ref | Expression |
---|---|
evenprm2 | ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2a1 28 | . . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2))) | |
2 | df-ne 3000 | . . . . . . . . 9 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2) | |
3 | 2 | biimpri 220 | . . . . . . . 8 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2) |
4 | 3 | anim2i 610 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
5 | 4 | ancoms 452 | . . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
6 | eldifsn 4538 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
7 | 5, 6 | sylibr 226 | . . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℙ ∖ {2})) |
8 | oddprmALTV 42442 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd ) | |
9 | oddneven 42401 | . . . . . 6 ⊢ (𝑃 ∈ Odd → ¬ 𝑃 ∈ Even ) | |
10 | 9 | pm2.21d 119 | . . . . 5 ⊢ (𝑃 ∈ Odd → (𝑃 ∈ Even → 𝑃 = 2)) |
11 | 7, 8, 10 | 3syl 18 | . . . 4 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ Even → 𝑃 = 2)) |
12 | 11 | ex 403 | . . 3 ⊢ (¬ 𝑃 = 2 → (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2))) |
13 | 1, 12 | pm2.61i 177 | . 2 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2)) |
14 | 2evenALTV 42447 | . . 3 ⊢ 2 ∈ Even | |
15 | eleq1 2894 | . . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ Even ↔ 2 ∈ Even )) | |
16 | 14, 15 | mpbiri 250 | . 2 ⊢ (𝑃 = 2 → 𝑃 ∈ Even ) |
17 | 13, 16 | impbid1 217 | 1 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∖ cdif 3795 {csn 4399 2c2 11413 ℙcprime 15764 Even ceven 42381 Odd codd 42382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-dvds 15365 df-prm 15765 df-even 42383 df-odd 42384 |
This theorem is referenced by: oddprmne2 42468 sbgoldbaltlem1 42511 |
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