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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2nodd | Structured version Visualization version GIF version |
Description: 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.) |
Ref | Expression |
---|---|
oddinmgm.e | ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} |
Ref | Expression |
---|---|
2nodd | ⊢ 2 ∉ 𝑂 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfnz 12637 | . . . . . . . . 9 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | eleq1 2822 | . . . . . . . . 9 ⊢ ((1 / 2) = 𝑥 → ((1 / 2) ∈ ℤ ↔ 𝑥 ∈ ℤ)) | |
3 | 1, 2 | mtbii 326 | . . . . . . . 8 ⊢ ((1 / 2) = 𝑥 → ¬ 𝑥 ∈ ℤ) |
4 | 3 | con2i 139 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ¬ (1 / 2) = 𝑥) |
5 | 1cnd 11206 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 1 ∈ ℂ) | |
6 | zcn 12560 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
7 | 2cnd 12287 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ∈ ℂ) | |
8 | 2ne0 12313 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ≠ 0) |
10 | 5, 6, 7, 9 | divmul2d 12020 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) |
11 | 4, 10 | mtbid 324 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ 1 = (2 · 𝑥)) |
12 | eqcom 2740 | . . . . . . . 8 ⊢ (2 = ((2 · 𝑥) + 1) ↔ ((2 · 𝑥) + 1) = 2) | |
13 | 12 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (2 = ((2 · 𝑥) + 1) ↔ ((2 · 𝑥) + 1) = 2)) |
14 | 7, 6 | mulcld 11231 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 · 𝑥) ∈ ℂ) |
15 | subadd2 11461 | . . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → ((2 − 1) = (2 · 𝑥) ↔ ((2 · 𝑥) + 1) = 2)) | |
16 | 15 | bicomd 222 | . . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
17 | 7, 5, 14, 16 | syl3anc 1372 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
18 | 2m1e1 12335 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 − 1) = 1) |
20 | 19 | eqeq1d 2735 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((2 − 1) = (2 · 𝑥) ↔ 1 = (2 · 𝑥))) |
21 | 13, 17, 20 | 3bitrd 305 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (2 = ((2 · 𝑥) + 1) ↔ 1 = (2 · 𝑥))) |
22 | 11, 21 | mtbird 325 | . . . . 5 ⊢ (𝑥 ∈ ℤ → ¬ 2 = ((2 · 𝑥) + 1)) |
23 | 22 | nrex 3075 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1) |
24 | 23 | intnan 488 | . . 3 ⊢ ¬ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1)) |
25 | eqeq1 2737 | . . . . 5 ⊢ (𝑧 = 2 → (𝑧 = ((2 · 𝑥) + 1) ↔ 2 = ((2 · 𝑥) + 1))) | |
26 | 25 | rexbidv 3179 | . . . 4 ⊢ (𝑧 = 2 → (∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1) ↔ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1))) |
27 | oddinmgm.e | . . . 4 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} | |
28 | 26, 27 | elrab2 3686 | . . 3 ⊢ (2 ∈ 𝑂 ↔ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1))) |
29 | 24, 28 | mtbir 323 | . 2 ⊢ ¬ 2 ∈ 𝑂 |
30 | 29 | nelir 3050 | 1 ⊢ 2 ∉ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∉ wnel 3047 ∃wrex 3071 {crab 3433 (class class class)co 7406 ℂcc 11105 0cc0 11107 1c1 11108 + caddc 11110 · cmul 11112 − cmin 11441 / cdiv 11868 2c2 12264 ℤcz 12555 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 |
This theorem is referenced by: oddinmgm 46572 |
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