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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 2944 | . . . . . . . . 9 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2) | |
3 | 2 | biimpri 227 | . . . . . . . 8 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2) |
4 | 3 | anim2i 617 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
5 | 4 | ancoms 459 | . . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
6 | eldifsn 4721 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
7 | 5, 6 | sylibr 233 | . . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℙ ∖ {2})) |
8 | oddprmALTV 45095 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd ) | |
9 | oddneven 45052 | . . . . . 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 413 | . . 3 ⊢ (¬ 𝑃 = 2 → (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2))) |
13 | 1, 12 | pm2.61i 182 | . 2 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2)) |
14 | 2evenALTV 45100 | . . 3 ⊢ 2 ∈ Even | |
15 | eleq1 2826 | . . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ Even ↔ 2 ∈ Even )) | |
16 | 14, 15 | mpbiri 257 | . 2 ⊢ (𝑃 = 2 → 𝑃 ∈ Even ) |
17 | 13, 16 | impbid1 224 | 1 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 {csn 4562 2c2 12016 ℙcprime 16364 Even ceven 45032 Odd codd 45033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-sup 9189 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-rp 12719 df-seq 13710 df-exp 13771 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-dvds 15952 df-prm 16365 df-even 45034 df-odd 45035 |
This theorem is referenced by: oddprmne2 45123 sbgoldbaltlem1 45187 |
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