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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 9147 | . 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 2141 class class class wbr 3987 0cc0 7761 ≤ cle 7942 ℕ0cn0 9122 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-iota 5158 df-fv 5204 df-ov 5853 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-inn 8866 df-n0 9123 |
This theorem is referenced by: flqmulnn0 10242 zmodfz 10289 addmodid 10315 modifeq2int 10329 modaddmodlo 10331 modsumfzodifsn 10339 addmodlteq 10341 expnnval 10466 nn0le2msqd 10640 facwordi 10661 faclbnd 10662 faclbnd6 10665 facavg 10667 geolim2 11462 mertenslemi1 11485 eftabs 11606 efcllemp 11608 efaddlem 11624 eftlub 11640 oexpneg 11823 divalglemnn 11864 divalglemnqt 11866 divalglemeunn 11867 divalg2 11872 dfgcd2 11956 gcdmultiple 11962 gcdmultiplez 11963 dvdssqlem 11972 nn0seqcvgd 11982 mulgcddvds 12035 isprm5lem 12082 nn0sqrtelqelz 12147 nonsq 12148 phibndlem 12157 dfphi2 12161 modprm0 12195 pythagtriplem3 12208 pythagtriplem10 12210 pythagtriplem6 12211 pythagtriplem7 12212 pythagtriplem12 12216 pythagtriplem14 12218 pcge0 12253 pcprmpw2 12273 pcmptdvds 12284 fldivp1 12287 pcbc 12290 qexpz 12291 pockthlem 12295 pockthg 12296 mul4sqlem 12332 ennnfoneleminc 12353 logbgcd1irraplemexp 13639 lgsval2lem 13664 lgsval4a 13676 2sqlem3 13706 2sqlem7 13710 2sqlem8 13712 |
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