Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 11455 | . . . . . . . . 9 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | eleq1 2689 | . . . . . . . . 9 ⊢ ((1 / 2) = 𝑥 → ((1 / 2) ∈ ℤ ↔ 𝑥 ∈ ℤ)) | |
3 | 1, 2 | mtbii 316 | . . . . . . . 8 ⊢ ((1 / 2) = 𝑥 → ¬ 𝑥 ∈ ℤ) |
4 | 3 | con2i 134 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ¬ (1 / 2) = 𝑥) |
5 | 1cnd 10056 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 1 ∈ ℂ) | |
6 | zcn 11382 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
7 | 2cnd 11093 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ∈ ℂ) | |
8 | 2ne0 11113 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 2 ≠ 0) |
10 | 5, 6, 7, 9 | divmul2d 10834 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) |
11 | 4, 10 | mtbid 314 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ 1 = (2 · 𝑥)) |
12 | eqcom 2629 | . . . . . . . 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 10060 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 · 𝑥) ∈ ℂ) |
15 | subadd2 10285 | . . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → ((2 − 1) = (2 · 𝑥) ↔ ((2 · 𝑥) + 1) = 2)) | |
16 | 15 | bicomd 213 | . . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (2 · 𝑥) ∈ ℂ) → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
17 | 7, 5, 14, 16 | syl3anc 1326 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (((2 · 𝑥) + 1) = 2 ↔ (2 − 1) = (2 · 𝑥))) |
18 | 2m1e1 11135 | . . . . . . . . 9 ⊢ (2 − 1) = 1 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → (2 − 1) = 1) |
20 | 19 | eqeq1d 2624 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((2 − 1) = (2 · 𝑥) ↔ 1 = (2 · 𝑥))) |
21 | 13, 17, 20 | 3bitrd 294 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (2 = ((2 · 𝑥) + 1) ↔ 1 = (2 · 𝑥))) |
22 | 11, 21 | mtbird 315 | . . . . 5 ⊢ (𝑥 ∈ ℤ → ¬ 2 = ((2 · 𝑥) + 1)) |
23 | 22 | nrex 3000 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1) |
24 | 23 | intnan 960 | . . 3 ⊢ ¬ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1)) |
25 | eqeq1 2626 | . . . . 5 ⊢ (𝑧 = 2 → (𝑧 = ((2 · 𝑥) + 1) ↔ 2 = ((2 · 𝑥) + 1))) | |
26 | 25 | rexbidv 3052 | . . . 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 3366 | . . 3 ⊢ (2 ∈ 𝑂 ↔ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = ((2 · 𝑥) + 1))) |
29 | 24, 28 | mtbir 313 | . 2 ⊢ ¬ 2 ∈ 𝑂 |
30 | 29 | nelir 2900 | 1 ⊢ 2 ∉ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∃wrex 2913 {crab 2916 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 / cdiv 10684 2c2 11070 ℤcz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 |
This theorem is referenced by: oddinmgm 41815 |
Copyright terms: Public domain | W3C validator |