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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 12063 | . . . . . . . . 9 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | eleq1 2902 | . . . . . . . . 9 ⊢ ((1 / 2) = 𝑥 → ((1 / 2) ∈ ℤ ↔ 𝑥 ∈ ℤ)) | |
3 | 1, 2 | mtbii 328 | . . . . . . . 8 ⊢ ((1 / 2) = 𝑥 → ¬ 𝑥 ∈ ℤ) |
4 | 3 | con2i 141 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ¬ (1 / 2) = 𝑥) |
5 | 1cnd 10638 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 1 ∈ ℂ) | |
6 | zcn 11989 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
7 | 2cnd 11718 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ∈ ℂ) | |
8 | 2ne0 11744 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ≠ 0) |
10 | 5, 6, 7, 9 | divmul2d 11451 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) |
11 | 4, 10 | mtbid 326 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ 1 = (2 · 𝑥)) |
12 | eqcom 2830 | . . . . . . . 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 10663 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 · 𝑥) ∈ ℂ) |
15 | subadd2 10892 | . . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → ((2 − 1) = (2 · 𝑥) ↔ ((2 · 𝑥) + 1) = 2)) | |
16 | 15 | bicomd 225 | . . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
17 | 7, 5, 14, 16 | syl3anc 1367 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
18 | 2m1e1 11766 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 − 1) = 1) |
20 | 19 | eqeq1d 2825 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((2 − 1) = (2 · 𝑥) ↔ 1 = (2 · 𝑥))) |
21 | 13, 17, 20 | 3bitrd 307 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (2 = ((2 · 𝑥) + 1) ↔ 1 = (2 · 𝑥))) |
22 | 11, 21 | mtbird 327 | . . . . 5 ⊢ (𝑥 ∈ ℤ → ¬ 2 = ((2 · 𝑥) + 1)) |
23 | 22 | nrex 3271 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1) |
24 | 23 | intnan 489 | . . 3 ⊢ ¬ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1)) |
25 | eqeq1 2827 | . . . . 5 ⊢ (𝑧 = 2 → (𝑧 = ((2 · 𝑥) + 1) ↔ 2 = ((2 · 𝑥) + 1))) | |
26 | 25 | rexbidv 3299 | . . . 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 3685 | . . 3 ⊢ (2 ∈ 𝑂 ↔ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1))) |
29 | 24, 28 | mtbir 325 | . 2 ⊢ ¬ 2 ∈ 𝑂 |
30 | 29 | nelir 3128 | 1 ⊢ 2 ∉ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∉ wnel 3125 ∃wrex 3141 {crab 3144 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 − cmin 10872 / cdiv 11299 2c2 11695 ℤcz 11984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 |
This theorem is referenced by: oddinmgm 44089 |
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