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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 2736 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 = 0 ↔ 𝑁 = 0)) | |
2 | eleq1 2821 | . . 3 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ ℕ ↔ 𝑁 ∈ ℕ)) | |
3 | negeq 11448 | . . . 4 ⊢ (𝑥 = 𝑁 → -𝑥 = -𝑁) | |
4 | 3 | eleq1d 2818 | . . 3 ⊢ (𝑥 = 𝑁 → (-𝑥 ∈ ℕ ↔ -𝑁 ∈ ℕ)) |
5 | 1, 2, 4 | 3orbi123d 1435 | . 2 ⊢ (𝑥 = 𝑁 → ((𝑥 = 0 ∨ 𝑥 ∈ ℕ ∨ -𝑥 ∈ ℕ) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
6 | df-z 12555 | . 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 205 ∧ wa 396 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ℝcr 11105 0cc0 11106 -cneg 11441 ℕcn 12208 ℤcz 12554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7408 df-neg 11443 df-z 12555 |
This theorem is referenced by: nnnegz 12557 zre 12558 elnnz 12564 0z 12565 elznn0nn 12568 elznn0 12569 elznn 12570 nnz 12575 znegcl 12593 zeo 12644 addmodlteq 13907 zabsle1 26788 ostthlem1 27119 ostth3 27130 elzdif0 32948 qqhval2lem 32949 exp11d 41211 |
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