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Mirrors > Home > ILE Home > Th. List > elnn0 | GIF version |
Description: Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.) |
Ref | Expression |
---|---|
elnn0 | ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 7958 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | eleq2i 2101 | . 2 ⊢ (A ∈ ℕ0 ↔ A ∈ (ℕ ∪ {0})) |
3 | elun 3078 | . 2 ⊢ (A ∈ (ℕ ∪ {0}) ↔ (A ∈ ℕ ∨ A ∈ {0})) | |
4 | c0ex 6819 | . . . 4 ⊢ 0 ∈ V | |
5 | 4 | elsnc2 3397 | . . 3 ⊢ (A ∈ {0} ↔ A = 0) |
6 | 5 | orbi2i 678 | . 2 ⊢ ((A ∈ ℕ ∨ A ∈ {0}) ↔ (A ∈ ℕ ∨ A = 0)) |
7 | 2, 3, 6 | 3bitri 195 | 1 ⊢ (A ∈ ℕ0 ↔ (A ∈ ℕ ∨ A = 0)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ∪ cun 2909 {csn 3367 0cc0 6711 ℕcn 7695 ℕ0cn0 7957 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-1cn 6776 ax-icn 6778 ax-addcl 6779 ax-mulcl 6781 ax-i2m1 6788 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-n0 7958 |
This theorem is referenced by: 0nn0 7972 nn0ge0 7983 nnnn0addcl 7988 nnm1nn0 7999 elnnnn0b 8002 elnn0z 8034 elznn0nn 8035 elznn0 8036 elznn 8037 nn0ind-raph 8131 expp1 8916 expnegap0 8917 expcllem 8920 |
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