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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 8735 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnex 8489 | . . 3 ⊢ ℕ ∈ V | |
3 | c0ex 7543 | . . . 4 ⊢ 0 ∈ V | |
4 | 3 | snex 4026 | . . 3 ⊢ {0} ∈ V |
5 | 2, 4 | unex 4276 | . 2 ⊢ (ℕ ∪ {0}) ∈ V |
6 | 1, 5 | eqeltri 2161 | 1 ⊢ ℕ0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 Vcvv 2620 ∪ cun 2998 {csn 3450 0cc0 7411 ℕcn 8483 ℕ0cn0 8734 |
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 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-i2m1 7511 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-uni 3660 df-int 3695 df-inn 8484 df-n0 8735 |
This theorem is referenced by: nn0ennn 9901 nnenom 9902 expcnvap0 10957 expcnvre 10958 expcnv 10959 geolim 10966 mertenslem2 10991 eftlub 11041 znnen 11550 |
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