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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 12220 | . . . . . . . . 9 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | eleq1 2818 | . . . . . . . . 9 ⊢ ((1 / 2) = 𝑥 → ((1 / 2) ∈ ℤ ↔ 𝑥 ∈ ℤ)) | |
3 | 1, 2 | mtbii 329 | . . . . . . . 8 ⊢ ((1 / 2) = 𝑥 → ¬ 𝑥 ∈ ℤ) |
4 | 3 | con2i 141 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ¬ (1 / 2) = 𝑥) |
5 | 1cnd 10793 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 1 ∈ ℂ) | |
6 | zcn 12146 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
7 | 2cnd 11873 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ∈ ℂ) | |
8 | 2ne0 11899 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ≠ 0) |
10 | 5, 6, 7, 9 | divmul2d 11606 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) |
11 | 4, 10 | mtbid 327 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ 1 = (2 · 𝑥)) |
12 | eqcom 2743 | . . . . . . . 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 10818 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 · 𝑥) ∈ ℂ) |
15 | subadd2 11047 | . . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → ((2 − 1) = (2 · 𝑥) ↔ ((2 · 𝑥) + 1) = 2)) | |
16 | 15 | bicomd 226 | . . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
17 | 7, 5, 14, 16 | syl3anc 1373 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
18 | 2m1e1 11921 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 − 1) = 1) |
20 | 19 | eqeq1d 2738 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((2 − 1) = (2 · 𝑥) ↔ 1 = (2 · 𝑥))) |
21 | 13, 17, 20 | 3bitrd 308 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (2 = ((2 · 𝑥) + 1) ↔ 1 = (2 · 𝑥))) |
22 | 11, 21 | mtbird 328 | . . . . 5 ⊢ (𝑥 ∈ ℤ → ¬ 2 = ((2 · 𝑥) + 1)) |
23 | 22 | nrex 3178 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1) |
24 | 23 | intnan 490 | . . 3 ⊢ ¬ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1)) |
25 | eqeq1 2740 | . . . . 5 ⊢ (𝑧 = 2 → (𝑧 = ((2 · 𝑥) + 1) ↔ 2 = ((2 · 𝑥) + 1))) | |
26 | 25 | rexbidv 3206 | . . . 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 3594 | . . 3 ⊢ (2 ∈ 𝑂 ↔ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1))) |
29 | 24, 28 | mtbir 326 | . 2 ⊢ ¬ 2 ∈ 𝑂 |
30 | 29 | nelir 3039 | 1 ⊢ 2 ∉ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∉ wnel 3036 ∃wrex 3052 {crab 3055 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 − cmin 11027 / cdiv 11454 2c2 11850 ℤcz 12141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 |
This theorem is referenced by: oddinmgm 44985 |
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