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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 8906 | . 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 1463 class class class wbr 3895 0cc0 7547 ≤ cle 7725 ℕ0cn0 8881 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-i2m1 7650 ax-0lt1 7651 ax-0id 7653 ax-rnegex 7654 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-xp 4505 df-cnv 4507 df-iota 5046 df-fv 5089 df-ov 5731 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-inn 8631 df-n0 8882 |
This theorem is referenced by: flqmulnn0 9965 zmodfz 10012 addmodid 10038 modifeq2int 10052 modaddmodlo 10054 modsumfzodifsn 10062 addmodlteq 10064 expnnval 10189 nn0le2msqd 10358 facwordi 10379 faclbnd 10380 faclbnd6 10383 facavg 10385 geolim2 11173 mertenslemi1 11196 eftabs 11213 efcllemp 11215 efaddlem 11231 eftlub 11247 oexpneg 11422 divalglemnn 11463 divalglemnqt 11465 divalglemeunn 11466 divalg2 11471 dfgcd2 11548 gcdmultiple 11554 gcdmultiplez 11555 dvdssqlem 11564 nn0seqcvgd 11568 mulgcddvds 11621 nn0sqrtelqelz 11729 nonsq 11730 phibndlem 11737 dfphi2 11741 ennnfoneleminc 11769 |
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