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Mirrors > Home > MPE Home > Th. List > elz | Structured version Visualization version GIF version |
Description: Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
elz | ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2827 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0)) | |
2 | eleq1 2902 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
3 | negeq 10880 | . . . 4 ⊢ (𝑥 = 𝑁 → -𝑥 = -𝑁) | |
4 | 3 | eleq1d 2899 | . . 3 ⊢ (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ)) |
5 | 1, 2, 4 | 3orbi123d 1431 | . 2 ⊢ (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
6 | df-z 11985 | . 2 ⊢ ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)} | |
7 | 5, 6 | elrab2 3685 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ℝcr 10538 0cc0 10539 -cneg 10873 ℕcn 11640 ℤcz 11984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-neg 10875 df-z 11985 |
This theorem is referenced by: nnnegz 11987 zre 11988 elnnz 11994 0z 11995 elznn0nn 11998 elznn0 11999 elznn 12000 nnssz 12005 znegcl 12020 zeo 12071 addmodlteq 13317 zabsle1 25874 ostthlem1 26205 ostth3 26216 elzdif0 31223 qqhval2lem 31224 |
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