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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 9107 | . 2 ⊢ ℕ0 = (ℕ ∪ {0}) | |
2 | nnex 8855 | . . 3 ⊢ ℕ ∈ V | |
3 | c0ex 7885 | . . . 4 ⊢ 0 ∈ V | |
4 | 3 | snex 4159 | . . 3 ⊢ {0} ∈ V |
5 | 2, 4 | unex 4414 | . 2 ⊢ (ℕ ∪ {0}) ∈ V |
6 | 1, 5 | eqeltri 2237 | 1 ⊢ ℕ0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 Vcvv 2722 ∪ cun 3110 {csn 3571 0cc0 7745 ℕcn 8849 ℕ0cn0 9106 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-i2m1 7850 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-uni 3785 df-int 3820 df-inn 8850 df-n0 9107 |
This theorem is referenced by: nn0ennn 10359 nnenom 10360 uzennn 10362 expcnvap0 11433 expcnvre 11434 expcnv 11435 geolim 11442 mertenslem2 11467 eftlub 11621 1arith 12286 znnen 12294 |
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