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Mirrors > Home > ILE Home > Th. List > elz | 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 2087 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0)) | |
2 | eleq1 2141 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
3 | negeq 7301 | . . . 4 ⊢ (𝑥 = 𝑁 → -𝑥 = -𝑁) | |
4 | 3 | eleq1d 2147 | . . 3 ⊢ (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ)) |
5 | 1, 2, 4 | 3orbi123d 1242 | . 2 ⊢ (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
6 | df-z 8352 | . 2 ⊢ ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)} | |
7 | 5, 6 | elrab2 2751 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∨ w3o 918 = wceq 1284 ∈ wcel 1433 ℝcr 6980 0cc0 6981 -cneg 7280 ℕcn 8039 ℤcz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-rab 2357 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 df-neg 7282 df-z 8352 |
This theorem is referenced by: nnnegz 8354 zre 8355 elnnz 8361 0z 8362 elnn0z 8364 elznn0nn 8365 elznn0 8366 elznn 8367 znegcl 8382 zaddcl 8391 ztri3or0 8393 zeo 8452 addmodlteq 9400 |
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