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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 9232 | . 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 2160 class class class wbr 4018 0cc0 7842 ≤ cle 8024 ℕ0cn0 9207 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-iota 5196 df-fv 5243 df-ov 5900 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-inn 8951 df-n0 9208 |
This theorem is referenced by: flqmulnn0 10332 zmodfz 10379 addmodid 10405 modifeq2int 10419 modaddmodlo 10421 modsumfzodifsn 10429 addmodlteq 10431 expnnval 10557 nn0le2msqd 10734 facwordi 10755 faclbnd 10756 faclbnd6 10759 facavg 10761 geolim2 11555 mertenslemi1 11578 eftabs 11699 efcllemp 11701 efaddlem 11717 eftlub 11733 oexpneg 11917 divalglemnn 11958 divalglemnqt 11960 divalglemeunn 11961 divalg2 11966 dfgcd2 12050 gcdmultiple 12056 gcdmultiplez 12057 dvdssqlem 12066 nn0seqcvgd 12076 mulgcddvds 12129 isprm5lem 12176 nn0sqrtelqelz 12241 nonsq 12242 phibndlem 12251 dfphi2 12255 modprm0 12289 pythagtriplem3 12302 pythagtriplem10 12304 pythagtriplem6 12305 pythagtriplem7 12306 pythagtriplem12 12310 pythagtriplem14 12312 pcge0 12348 pcprmpw2 12368 pcmptdvds 12380 fldivp1 12383 pcbc 12386 qexpz 12387 pockthlem 12391 pockthg 12392 mul4sqlem 12428 4sqlem12 12437 4sqlem14 12439 4sqlem16 12441 ennnfoneleminc 12465 logbgcd1irraplemexp 14863 wilthlem1 14875 lgsval2lem 14889 lgsval4a 14901 lgseisenlem1 14928 lgseisenlem2 14929 2sqlem3 14942 2sqlem7 14946 2sqlem8 14948 |
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