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Mirrors > Home > ILE Home > Th. List > nn0ge0d | GIF version |
Description: A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0ge0d | ⊢ (𝜑 → 0 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | nn0ge0 8313 | . 2 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 0 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 class class class wbr 3785 0cc0 6981 ≤ cle 7154 ℕ0cn0 8288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-iota 4887 df-fv 4930 df-ov 5535 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-inn 8040 df-n0 8289 |
This theorem is referenced by: flqmulnn0 9301 zmodfz 9348 addmodid 9374 modifeq2int 9388 modaddmodlo 9390 modsumfzodifsn 9398 addmodlteq 9400 expinnval 9479 nn0le2msqd 9646 facwordi 9667 faclbnd 9668 faclbnd6 9671 facavg 9673 oexpneg 10276 divalglemnn 10318 divalglemnqt 10320 divalglemeunn 10321 divalg2 10326 dfgcd2 10403 gcdmultiple 10409 gcdmultiplez 10410 dvdssqlem 10419 nn0seqcvgd 10423 mulgcddvds 10476 |
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