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Mirrors > Home > ILE Home > Th. List > nnge1d | GIF version |
Description: A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnge1d | ⊢ (𝜑 → 1 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | nnge1 8743 | . 2 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 1 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 1c1 7621 ≤ cle 7801 ℕcn 8720 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 |
This theorem is referenced by: exbtwnzlemstep 10025 addmodlteq 10171 bernneq3 10414 facwordi 10486 faclbnd 10487 faclbnd3 10489 facavg 10492 bcval5 10509 1elfz0hash 10552 seq3coll 10585 fsumcl2lem 11167 eftlub 11396 eflegeo 11408 eirraplem 11483 divdenle 11875 |
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