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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 12588 | . 2 ⊢ 1 ∈ ℤ | |
2 | 0z 12565 | . . 3 ⊢ 0 ∈ ℤ | |
3 | id 22 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℤ) | |
4 | oveq2 7412 | . . . . . . . 8 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 2t0e0 12377 | . . . . . . . 8 ⊢ (2 · 0) = 0 | |
6 | 4, 5 | eqtrdi 2789 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0) |
7 | 6 | oveq1d 7419 | . . . . . 6 ⊢ (𝑥 = 0 → ((2 · 𝑥) + 1) = (0 + 1)) |
8 | 7 | eqeq2d 2744 | . . . . 5 ⊢ (𝑥 = 0 → (1 = ((2 · 𝑥) + 1) ↔ 1 = (0 + 1))) |
9 | 8 | adantl 483 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑥 = 0) → (1 = ((2 · 𝑥) + 1) ↔ 1 = (0 + 1))) |
10 | 1e0p1 12715 | . . . . 5 ⊢ 1 = (0 + 1) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (0 ∈ ℤ → 1 = (0 + 1)) |
12 | 3, 9, 11 | rspcedvd 3614 | . . 3 ⊢ (0 ∈ ℤ → ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1)) |
13 | 2, 12 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1) |
14 | eqeq1 2737 | . . . 4 ⊢ (𝑧 = 1 → (𝑧 = ((2 · 𝑥) + 1) ↔ 1 = ((2 · 𝑥) + 1))) | |
15 | 14 | rexbidv 3179 | . . 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 3685 | . 2 ⊢ (1 ∈ 𝑂 ↔ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1))) |
18 | 1, 13, 17 | mpbir2an 710 | 1 ⊢ 1 ∈ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 (class class class)co 7404 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 2c2 12263 ℤcz 12554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-neg 11443 df-nn 12209 df-2 12271 df-z 12555 |
This theorem is referenced by: oddinmgm 46520 |
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