![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2739 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0)) | |
2 | eleq1 2827 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
3 | negeq 11498 | . . . 4 ⊢ (𝑥 = 𝑁 → -𝑥 = -𝑁) | |
4 | 3 | eleq1d 2824 | . . 3 ⊢ (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ)) |
5 | 1, 2, 4 | 3orbi123d 1434 | . 2 ⊢ (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
6 | df-z 12612 | . 2 ⊢ ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)} | |
7 | 5, 6 | elrab2 3698 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∨ w3o 1085 = wceq 1537 ∈ wcel 2106 ℝcr 11152 0cc0 11153 -cneg 11491 ℕcn 12264 ℤcz 12611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-neg 11493 df-z 12612 |
This theorem is referenced by: nnnegz 12614 zre 12615 elnnz 12621 0z 12622 elznn0nn 12625 elznn0 12626 elznn 12627 nnz 12632 znegcl 12650 zeo 12702 addmodlteq 13984 zabsle1 27355 ostthlem1 27686 ostth3 27697 elzdif0 33943 qqhval2lem 33944 exp11d 42340 |
Copyright terms: Public domain | W3C validator |