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Mirrors > Home > ILE Home > Th. List > nnne0i | GIF version |
Description: A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
nngt0.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
nnne0i | ⊢ 𝐴 ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nngt0.1 | . . 3 ⊢ 𝐴 ∈ ℕ | |
2 | 1 | nnrei 8926 | . 2 ⊢ 𝐴 ∈ ℝ |
3 | 1 | nngt0i 8947 | . 2 ⊢ 0 < 𝐴 |
4 | 2, 3 | gt0ne0ii 8442 | 1 ⊢ 𝐴 ≠ 0 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 ≠ wne 2347 0cc0 7810 ℕcn 8917 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-iota 5178 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-inn 8918 |
This theorem is referenced by: 3lcm2e6woprm 12080 6lcm4e12 12081 |
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