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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 11284 | . 2 ⊢ 1 ∈ ℤ | |
2 | 0z 11265 | . . 3 ⊢ 0 ∈ ℤ | |
3 | id 22 | . . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℤ) | |
4 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 2t0e0 11060 | . . . . . . . 8 ⊢ (2 · 0) = 0 | |
6 | 4, 5 | syl6eq 2660 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0) |
7 | 6 | oveq1d 6564 | . . . . . 6 ⊢ (𝑥 = 0 → ((2 · 𝑥) + 1) = (0 + 1)) |
8 | 7 | eqeq2d 2620 | . . . . 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 11428 | . . . . 5 ⊢ 1 = (0 + 1) | |
11 | 10 | a1i 11 | . . . 4 ⊢ (0 ∈ ℤ → 1 = (0 + 1)) |
12 | 3, 9, 11 | rspcedvd 3289 | . . 3 ⊢ (0 ∈ ℤ → ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1)) |
13 | 2, 12 | ax-mp 5 | . 2 ⊢ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1) |
14 | eqeq1 2614 | . . . 4 ⊢ (𝑧 = 1 → (𝑧 = ((2 · 𝑥) + 1) ↔ 1 = ((2 · 𝑥) + 1))) | |
15 | 14 | rexbidv 3034 | . . 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 3333 | . 2 ⊢ (1 ∈ 𝑂 ↔ (1 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 1 = ((2 · 𝑥) + 1))) |
18 | 1, 13, 17 | mpbir2an 957 | 1 ⊢ 1 ∈ 𝑂 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 2c2 10947 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-z 11255 |
This theorem is referenced by: oddinmgm 41605 |
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