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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 3017 | . . . . . . . . 9 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2) | |
3 | 2 | biimpri 230 | . . . . . . . 8 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2) |
4 | 3 | anim2i 618 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
5 | 4 | ancoms 461 | . . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
6 | eldifsn 4713 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
7 | 5, 6 | sylibr 236 | . . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℙ ∖ {2})) |
8 | oddprmALTV 43845 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd ) | |
9 | oddneven 43802 | . . . . . 6 ⊢ (𝑃 ∈ Odd → ¬ 𝑃 ∈ Even ) | |
10 | 9 | pm2.21d 121 | . . . . 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 415 | . . 3 ⊢ (¬ 𝑃 = 2 → (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2))) |
13 | 1, 12 | pm2.61i 184 | . 2 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2)) |
14 | 2evenALTV 43850 | . . 3 ⊢ 2 ∈ Even | |
15 | eleq1 2900 | . . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ Even ↔ 2 ∈ Even )) | |
16 | 14, 15 | mpbiri 260 | . 2 ⊢ (𝑃 = 2 → 𝑃 ∈ Even ) |
17 | 13, 16 | impbid1 227 | 1 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 {csn 4561 2c2 11686 ℙcprime 16009 Even ceven 43782 Odd codd 43783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-prm 16010 df-even 43784 df-odd 43785 |
This theorem is referenced by: oddprmne2 43873 sbgoldbaltlem1 43937 |
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