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Mirrors > Home > ILE Home > Th. List > nnnn0d | GIF version |
Description: A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnnn0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
Ref | Expression |
---|---|
nnnn0d | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssnn0 8973 | . 2 ⊢ ℕ ⊆ ℕ0 | |
2 | nnnn0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
3 | 1, 2 | sseldi 3090 | 1 ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℕcn 8713 ℕ0cn0 8970 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-n0 8971 |
This theorem is referenced by: nn0ge2m1nn0 9031 nnzd 9165 eluzge2nn0 9358 modsumfzodifsn 10162 addmodlteq 10164 expnnval 10289 expgt1 10324 expaddzaplem 10329 expaddzap 10330 expmulzap 10332 expnbnd 10408 facwordi 10479 faclbnd 10480 facavg 10485 bcm1k 10499 bcval5 10502 1elfz0hash 10545 resqrexlemnm 10783 resqrexlemcvg 10784 summodc 11145 zsumdc 11146 bcxmas 11251 geo2sum 11276 geo2lim 11278 geoisum1 11281 geoisum1c 11282 cvgratnnlembern 11285 cvgratnnlemsumlt 11290 cvgratnnlemfm 11291 mertenslemi1 11297 eftabs 11351 efcllemp 11353 eftlub 11385 eirraplem 11472 dvdsfac 11547 divalglemnqt 11606 divalglemeunn 11607 gcdval 11637 gcdcl 11644 dvdsgcdidd 11671 mulgcd 11693 rplpwr 11704 rppwr 11705 lcmcl 11742 lcmgcdnn 11752 nprmdvds1 11809 rpexp 11820 pw2dvdslemn 11832 sqpweven 11842 2sqpwodd 11843 nn0sqrtelqelz 11873 phiprmpw 11887 crth 11889 cvgcmp2nlemabs 13216 trilpolemlt1 13223 |
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