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Mirrors > Home > ILE Home > Th. List > nnex | GIF version |
Description: The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
nnex | ⊢ ℕ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 7712 | . 2 ⊢ ℂ ∈ V | |
2 | nnsscn 8693 | . 2 ⊢ ℕ ⊆ ℂ | |
3 | 1, 2 | ssexi 4036 | 1 ⊢ ℕ ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 ℂcc 7586 ℕcn 8688 |
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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 |
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-v 2662 df-in 3047 df-ss 3054 df-int 3742 df-inn 8689 |
This theorem is referenced by: nn0ex 8951 nn0ennn 10174 climrecvg1n 11085 climcvg1nlem 11086 divcnv 11234 trireciplem 11237 expcnvap0 11239 expcnv 11241 geo2lim 11253 prmex 11721 qnumval 11790 qdenval 11791 oddennn 11832 evenennn 11833 xpnnen 11834 znnen 11838 qnnen 11871 ndxarg 11909 trilpo 13163 |
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