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Mirrors > Home > ILE Home > Th. List > nn0ex | GIF version |
Description: The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
Ref | Expression |
---|---|
nn0ex | ⊢ ℕ0 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 8946 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnex 8694 | . . 3 ⊢ ℕ ∈ V | |
3 | c0ex 7728 | . . . 4 ⊢ 0 ∈ V | |
4 | 3 | snex 4079 | . . 3 ⊢ {0} ∈ V |
5 | 2, 4 | unex 4332 | . 2 ⊢ (ℕ ∪ {0}) ∈ V |
6 | 1, 5 | eqeltri 2190 | 1 ⊢ ℕ0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 ∪ cun 3039 {csn 3497 0cc0 7588 ℕcn 8688 ℕ0cn0 8945 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-i2m1 7693 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-inn 8689 df-n0 8946 |
This theorem is referenced by: nn0ennn 10174 nnenom 10175 uzennn 10177 expcnvap0 11239 expcnvre 11240 expcnv 11241 geolim 11248 mertenslem2 11273 eftlub 11323 znnen 11838 |
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