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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 9208 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnex 8956 | . . 3 ⊢ ℕ ∈ V | |
3 | c0ex 7982 | . . . 4 ⊢ 0 ∈ V | |
4 | 3 | snex 4203 | . . 3 ⊢ {0} ∈ V |
5 | 2, 4 | unex 4459 | . 2 ⊢ (ℕ ∪ {0}) ∈ V |
6 | 1, 5 | eqeltri 2262 | 1 ⊢ ℕ0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 {csn 3607 0cc0 7842 ℕcn 8950 ℕ0cn0 9207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-i2m1 7947 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-inn 8951 df-n0 9208 |
This theorem is referenced by: nn0ennn 10466 nnenom 10467 uzennn 10469 expcnvap0 11545 expcnvre 11546 expcnv 11547 geolim 11554 mertenslem2 11579 eftlub 11733 1arith 12402 znnen 12452 psrval 13961 fnpsr 13962 psrbag 13964 psrbasg 13968 psrelbas 13969 psrplusgg 13971 psraddcl 13973 |
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