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Mirrors > Home > ILE Home > Th. List > elznn0nn | GIF version |
Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
elznn0nn | ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9049 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | andi 807 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
3 | df-3or 963 | . . . 4 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ)) | |
4 | 3 | anbi2i 452 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ))) |
5 | nn0re 8979 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 5 | pm4.71ri 389 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0)) |
7 | elnn0 8972 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
8 | orcom 717 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) | |
9 | 7, 8 | bitri 183 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) |
10 | 9 | anbi2i 452 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ))) |
11 | 6, 10 | bitri 183 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ))) |
12 | 11 | orbi1i 752 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
13 | 2, 4, 12 | 3bitr4i 211 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
14 | 1, 13 | bitri 183 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∨ wo 697 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ℝcr 7612 0cc0 7613 -cneg 7927 ℕcn 8713 ℕ0cn0 8970 ℤcz 9047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-i2m1 7718 ax-rnegex 7722 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: peano2z 9083 zindd 9162 expcl2lemap 10298 mulexpzap 10326 expaddzap 10330 expmulzap 10332 absexpzap 10845 |
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