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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 12350 | . 2 ⊢ 1 ∈ ℤ | |
2 | 0z 12330 | . . 3 ⊢ 0 ∈ ℤ | |
3 | id 22 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℤ) | |
4 | oveq2 7283 | . . . . . . . 8 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 2t0e0 12142 | . . . . . . . 8 ⊢ (2 · 0) = 0 | |
6 | 4, 5 | eqtrdi 2794 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0) |
7 | 6 | oveq1d 7290 | . . . . . 6 ⊢ (𝑥 = 0 → ((2 · 𝑥) + 1) = (0 + 1)) |
8 | 7 | eqeq2d 2749 | . . . . 5 ⊢ (𝑥 = 0 → (1 = ((2 · 𝑥) + 1) ↔ 1 = (0 + 1))) |
9 | 8 | adantl 482 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑥 = 0) → (1 = ((2 · 𝑥) + 1) ↔ 1 = (0 + 1))) |
10 | 1e0p1 12479 | . . . . 5 ⊢ 1 = (0 + 1) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (0 ∈ ℤ → 1 = (0 + 1)) |
12 | 3, 9, 11 | rspcedvd 3563 | . . 3 ⊢ (0 ∈ ℤ → ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1)) |
13 | 2, 12 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1) |
14 | eqeq1 2742 | . . . 4 ⊢ (𝑧 = 1 → (𝑧 = ((2 · 𝑥) + 1) ↔ 1 = ((2 · 𝑥) + 1))) | |
15 | 14 | rexbidv 3226 | . . 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 3627 | . 2 ⊢ (1 ∈ 𝑂 ↔ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1))) |
18 | 1, 13, 17 | mpbir2an 708 | 1 ⊢ 1 ∈ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 {crab 3068 (class class class)co 7275 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 2c2 12028 ℤcz 12319 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-neg 11208 df-nn 11974 df-2 12036 df-z 12320 |
This theorem is referenced by: oddinmgm 45369 |
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