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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 2938 | . . . . . . . . 9 ⊢ (𝑃 ≠ 2 ↔ ¬ 𝑃 = 2) | |
3 | 2 | biimpri 227 | . . . . . . . 8 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2) |
4 | 3 | anim2i 616 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
5 | 4 | ancoms 458 | . . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) |
6 | eldifsn 4791 | . . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2)) | |
7 | 5, 6 | sylibr 233 | . . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ (ℙ ∖ {2})) |
8 | oddprmALTV 47027 | . . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ Odd ) | |
9 | oddneven 46984 | . . . . . 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 412 | . . 3 ⊢ (¬ 𝑃 = 2 → (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2))) |
13 | 1, 12 | pm2.61i 182 | . 2 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ Even → 𝑃 = 2)) |
14 | 2evenALTV 47032 | . . 3 ⊢ 2 ∈ Even | |
15 | eleq1 2817 | . . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ Even ↔ 2 ∈ Even )) | |
16 | 14, 15 | mpbiri 258 | . 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 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 {csn 4629 2c2 12297 ℙcprime 16641 Even ceven 46964 Odd codd 46965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-prm 16642 df-even 46966 df-odd 46967 |
This theorem is referenced by: oddprmne2 47055 sbgoldbaltlem1 47119 |
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