Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1neven | Structured version Visualization version GIF version |
Description: 1 is not an even integer. (Contributed by AV, 12-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
Ref | Expression |
---|---|
1neven | ⊢ 1 ∉ 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfnz 11455 | . . . . . . 7 ⊢ ¬ (1 / 2) ∈ ℤ | |
2 | eleq1a 2696 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 → (1 / 2) ∈ ℤ)) | |
3 | 1, 2 | mtoi 190 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ¬ (1 / 2) = 𝑥) |
4 | 1cnd 10056 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 1 ∈ ℂ) | |
5 | zcn 11382 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | 2cnne0 11242 | . . . . . . . 8 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → (2 ∈ ℂ ∧ 2 ≠ 0)) |
8 | divmul2 10689 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) | |
9 | 4, 5, 7, 8 | syl3anc 1326 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → ((1 / 2) = 𝑥 ↔ 1 = (2 · 𝑥))) |
10 | 3, 9 | mtbid 314 | . . . . 5 ⊢ (𝑥 ∈ ℤ → ¬ 1 = (2 · 𝑥)) |
11 | 10 | nrex 3000 | . . . 4 ⊢ ¬ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥) |
12 | 11 | intnan 960 | . . 3 ⊢ ¬ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥)) |
13 | eqeq1 2626 | . . . . 5 ⊢ (𝑧 = 1 → (𝑧 = (2 · 𝑥) ↔ 1 = (2 · 𝑥))) | |
14 | 13 | rexbidv 3052 | . . . 4 ⊢ (𝑧 = 1 → (∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥) ↔ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥))) |
15 | 2zrng.e | . . . 4 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
16 | 14, 15 | elrab2 3366 | . . 3 ⊢ (1 ∈ 𝐸 ↔ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = (2 · 𝑥))) |
17 | 12, 16 | mtbir 313 | . 2 ⊢ ¬ 1 ∈ 𝐸 |
18 | 17 | nelir 2900 | 1 ⊢ 1 ∉ 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∉ wnel 2897 ∃wrex 2913 {crab 2916 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 / 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: 2zrngnmlid 41949 2zrngnmrid 41950 |
Copyright terms: Public domain | W3C validator |