![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nnre | GIF version |
Description: A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
nnre | ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssre 8919 | . 2 ⊢ ℕ ⊆ ℝ | |
2 | 1 | sseli 3151 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ℝcr 7807 ℕcn 8915 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4120 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-in 3135 df-ss 3142 df-int 3845 df-inn 8916 |
This theorem is referenced by: nnrei 8924 peano2nn 8927 nn1suc 8934 nnge1 8938 nnle1eq1 8939 nngt0 8940 nnnlt1 8941 nnap0 8944 nn2ge 8948 nn1gt1 8949 nndivre 8951 nnrecgt0 8953 nnsub 8954 arch 9169 nnrecl 9170 bndndx 9171 nn0ge0 9197 0mnnnnn0 9204 nnnegz 9252 elnnz 9259 elz2 9320 gtndiv 9344 prime 9348 btwnz 9368 qre 9621 elpq 9644 elpqb 9645 nnrp 9659 nnledivrp 9762 fzo1fzo0n0 10178 elfzo0le 10180 fzonmapblen 10182 ubmelfzo 10195 fzonn0p1p1 10208 elfzom1p1elfzo 10209 ubmelm1fzo 10221 subfzo0 10237 adddivflid 10287 flltdivnn0lt 10299 intfracq 10315 flqdiv 10316 m1modnnsub1 10365 addmodid 10367 modfzo0difsn 10390 nnlesq 10618 facndiv 10712 faclbnd 10714 faclbnd3 10716 bcval5 10736 seq3coll 10815 caucvgre 10983 efaddlem 11675 nndivdvds 11796 nno 11903 nnoddm1d2 11907 divalglemnn 11915 divalg2 11923 ndvdsadd 11928 gcdmultiple 12013 gcdmultiplez 12014 gcdzeq 12015 sqgcd 12022 dvdssqlem 12023 lcmgcdlem 12069 coprmgcdb 12080 qredeq 12088 qredeu 12089 prmdvdsfz 12131 sqrt2irr 12154 divdenle 12189 phibndlem 12208 hashgcdlem 12230 oddprm 12251 pythagtriplem10 12261 pythagtriplem12 12267 pythagtriplem14 12269 pythagtriplem16 12271 pythagtriplem19 12274 pclemub 12279 pc2dvds 12321 pcmpt 12333 fldivp1 12338 pcbc 12341 infpnlem1 12349 oddennn 12385 exmidunben 12419 mulgnegnn 12925 lgsval4a 14294 |
Copyright terms: Public domain | W3C validator |