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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 9257 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | andi 818 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
3 | df-3or 979 | . . . 4 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ)) | |
4 | 3 | anbi2i 457 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) ∨ -𝑁 ∈ ℕ))) |
5 | nn0re 9187 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | 5 | pm4.71ri 392 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0)) |
7 | elnn0 9180 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
8 | orcom 728 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) | |
9 | 7, 8 | bitri 184 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) |
10 | 9 | anbi2i 457 | . . . . 5 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ))) |
11 | 6, 10 | bitri 184 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ))) |
12 | 11 | orbi1i 763 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
13 | 2, 4, 12 | 3bitr4i 212 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
14 | 1, 13 | bitri 184 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∨ wo 708 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ℝcr 7812 0cc0 7813 -cneg 8131 ℕcn 8921 ℕ0cn0 9178 ℤcz 9255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4123 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-i2m1 7918 ax-rnegex 7922 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-un 3135 df-in 3137 df-ss 3144 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5880 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 |
This theorem is referenced by: peano2z 9291 zindd 9373 expcl2lemap 10534 mulexpzap 10562 expaddzap 10566 expmulzap 10568 absexpzap 11091 pcid 12325 mulgsubcl 13002 mulgneg 13006 |
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