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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 8882 | . . 3 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | 1 | eleq2i 2181 | . 2 ⊢ (𝐴 ∈ ℕ0 ↔ 𝐴 ∈ (ℕ ∪ {0})) |
3 | elun 3183 | . 2 ⊢ (𝐴 ∈ (ℕ ∪ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 ∈ {0})) | |
4 | c0ex 7684 | . . . 4 ⊢ 0 ∈ V | |
5 | 4 | elsn2 3525 | . . 3 ⊢ (𝐴 ∈ {0} ↔ 𝐴 = 0) |
6 | 5 | orbi2i 734 | . 2 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 ∈ {0}) ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
7 | 2, 3, 6 | 3bitri 205 | 1 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 680 = wceq 1314 ∈ wcel 1463 ∪ cun 3035 {csn 3493 0cc0 7547 ℕcn 8630 ℕ0cn0 8881 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-1cn 7638 ax-icn 7640 ax-addcl 7641 ax-mulcl 7643 ax-i2m1 7650 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-sn 3499 df-n0 8882 |
This theorem is referenced by: 0nn0 8896 nn0ge0 8906 nnnn0addcl 8911 nnm1nn0 8922 elnnnn0b 8925 elnn0z 8971 elznn0nn 8972 elznn0 8973 elznn 8974 nn0ind-raph 9072 nn0ledivnn 9447 expp1 10193 expnegap0 10194 expcllem 10197 facp1 10369 faclbnd 10380 faclbnd3 10382 bcn1 10397 bcval5 10402 hashnncl 10435 fz1f1o 11036 arisum 11159 arisum2 11160 ef0lem 11217 nn0enne 11447 nn0o1gt2 11450 dfgcd2 11548 mulgcd 11550 eucalgf 11582 eucalginv 11583 prmdvdsexpr 11674 rpexp1i 11678 nn0gcdsq 11723 |
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