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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 8674 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnex 8428 | . . 3 ⊢ ℕ ∈ V | |
3 | c0ex 7482 | . . . 4 ⊢ 0 ∈ V | |
4 | 3 | snex 4020 | . . 3 ⊢ {0} ∈ V |
5 | 2, 4 | unex 4266 | . 2 ⊢ (ℕ ∪ {0}) ∈ V |
6 | 1, 5 | eqeltri 2160 | 1 ⊢ ℕ0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 Vcvv 2619 ∪ cun 2997 {csn 3446 0cc0 7350 ℕcn 8422 ℕ0cn0 8673 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-i2m1 7450 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-inn 8423 df-n0 8674 |
This theorem is referenced by: nn0ennn 9840 nnenom 9841 expcnvap0 10896 expcnvre 10897 expcnv 10898 geolim 10905 mertenslem2 10930 eftlub 10980 eucialgcvga 11318 eucialg 11319 znnen 11489 |
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