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Mirrors > Home > ILE Home > Th. List > 0nnn | GIF version |
Description: Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
0nnn | ⊢ ¬ 0 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 8071 | . 2 ⊢ 0 < 1 | |
2 | nnnlt1 8931 | . 2 ⊢ (0 ∈ ℕ → ¬ 0 < 1) | |
3 | 1, 2 | mt2 640 | 1 ⊢ ¬ 0 ∈ ℕ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2148 class class class wbr 4000 0cc0 7799 1c1 7800 < clt 7979 ℕcn 8905 |
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-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-pre-ltirr 7911 ax-pre-lttrn 7913 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-iota 5174 df-fv 5220 df-ov 5872 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 df-inn 8906 |
This theorem is referenced by: nnne0 8933 dfn2 9175 nn0enne 11887 exprmfct 12118 coprm 12124 |
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