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| 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 2741 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0)) | |
| 2 | eleq1 2829 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
| 3 | negeq 11500 | . . . 4 ⊢ (𝑥 = 𝑁 → -𝑥 = -𝑁) | |
| 4 | 3 | eleq1d 2826 | . . 3 ⊢ (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ)) | 
| 5 | 1, 2, 4 | 3orbi123d 1437 | . 2 ⊢ (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | 
| 6 | df-z 12614 | . 2 ⊢ ℤ = {𝑥 ∈ ℝ ∣ (𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ)} | |
| 7 | 5, 6 | elrab2 3695 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ℝcr 11154 0cc0 11155 -cneg 11493 ℕcn 12266 ℤcz 12613 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 df-z 12614 | 
| This theorem is referenced by: nnnegz 12616 zre 12617 elnnz 12623 0z 12624 elznn0nn 12627 elznn0 12628 elznn 12629 nnz 12634 znegcl 12652 zeo 12704 addmodlteq 13987 zabsle1 27340 ostthlem1 27671 ostth3 27682 elzdif0 33981 qqhval2lem 33982 exp11d 42361 | 
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