![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1odd | Structured version Visualization version GIF version |
Description: 1 is an odd integer. (Contributed by AV, 3-Feb-2020.) |
Ref | Expression |
---|---|
oddinmgm.e | ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} |
Ref | Expression |
---|---|
1odd | ⊢ 1 ∈ 𝑂 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 11445 | . 2 ⊢ 1 ∈ ℤ | |
2 | 0z 11426 | . . 3 ⊢ 0 ∈ ℤ | |
3 | id 22 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℤ) | |
4 | oveq2 6698 | . . . . . . . 8 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 2t0e0 11221 | . . . . . . . 8 ⊢ (2 · 0) = 0 | |
6 | 4, 5 | syl6eq 2701 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0) |
7 | 6 | oveq1d 6705 | . . . . . 6 ⊢ (𝑥 = 0 → ((2 · 𝑥) + 1) = (0 + 1)) |
8 | 7 | eqeq2d 2661 | . . . . 5 ⊢ (𝑥 = 0 → (1 = ((2 · 𝑥) + 1) ↔ 1 = (0 + 1))) |
9 | 8 | adantl 481 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑥 = 0) → (1 = ((2 · 𝑥) + 1) ↔ 1 = (0 + 1))) |
10 | 1e0p1 11590 | . . . . 5 ⊢ 1 = (0 + 1) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (0 ∈ ℤ → 1 = (0 + 1)) |
12 | 3, 9, 11 | rspcedvd 3348 | . . 3 ⊢ (0 ∈ ℤ → ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1)) |
13 | 2, 12 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1) |
14 | eqeq1 2655 | . . . 4 ⊢ (𝑧 = 1 → (𝑧 = ((2 · 𝑥) + 1) ↔ 1 = ((2 · 𝑥) + 1))) | |
15 | 14 | rexbidv 3081 | . . 3 ⊢ (𝑧 = 1 → (∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1) ↔ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1))) |
16 | oddinmgm.e | . . 3 ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)} | |
17 | 15, 16 | elrab2 3399 | . 2 ⊢ (1 ∈ 𝑂 ↔ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1))) |
18 | 1, 13, 17 | mpbir2an 975 | 1 ⊢ 1 ∈ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1523 ∈ wcel 2030 ∃wrex 2942 {crab 2945 (class class class)co 6690 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 2c2 11108 ℤcz 11415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-z 11416 |
This theorem is referenced by: oddinmgm 42140 |
Copyright terms: Public domain | W3C validator |