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Mirrors > Home > ILE Home > Th. List > nnne0 | GIF version |
Description: A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.) |
Ref | Expression |
---|---|
nnne0 | ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nnn 8771 | . . 3 ⊢ ¬ 0 ∈ ℕ | |
2 | eleq1 2203 | . . 3 ⊢ (𝐴 = 0 → (𝐴 ∈ ℕ ↔ 0 ∈ ℕ)) | |
3 | 1, 2 | mtbiri 665 | . 2 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ ℕ) |
4 | 3 | necon2ai 2363 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 0cc0 7644 ℕcn 8744 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-pre-ltirr 7756 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-inn 8745 |
This theorem is referenced by: nnne0d 8789 divfnzn 9440 qreccl 9461 fzo1fzo0n0 9991 expnnval 10327 expnegap0 10332 hashnncl 10574 ef0lem 11403 dvdsval3 11533 nndivdvds 11535 modmulconst 11561 dvdsdivcl 11584 divalg2 11659 ndvdssub 11663 nndvdslegcd 11690 divgcdz 11696 divgcdnn 11699 gcdzeq 11746 eucalgf 11772 eucalginv 11773 lcmgcdlem 11794 qredeu 11814 cncongr1 11820 cncongr2 11821 divnumden 11910 divdenle 11911 phimullem 11937 hashgcdlem 11939 ennnfonelemjn 11951 dvexp2 12884 |
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