Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nnq | GIF version |
Description: A positive integer is rational. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
nnq | ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnssq 9588 | . 2 ⊢ ℕ ⊆ ℚ | |
2 | 1 | sseli 3143 | 1 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2141 ℕcn 8878 ℚcq 9578 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
This theorem depends on definitions: df-bi 116 df-3or 974 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-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-z 9213 df-q 9579 |
This theorem is referenced by: flqdiv 10277 modqmulnn 10298 zmodcl 10300 zmodfz 10302 zmodid2 10308 m1modnnsub1 10326 addmodid 10328 modifeq2int 10342 modaddmodup 10343 modaddmodlo 10344 modsumfzodifsn 10352 addmodlteq 10354 modfsummodlemstep 11420 fprodmodd 11604 dvdsval3 11753 dvdsmodexp 11757 moddvds 11761 dvdslelemd 11803 dvdsmod 11822 mulmoddvds 11823 divalglemnn 11877 divalgmod 11886 modgcd 11946 crth 12178 phimullem 12179 eulerthlema 12184 fermltl 12188 prmdiv 12189 prmdiveq 12190 odzdvds 12199 modprm0 12208 nnnn0modprm0 12209 modprmn0modprm0 12210 pcaddlem 12292 fldivp1 12300 pockthlem 12308 pockthi 12310 4sqlem5 12334 4sqlem6 12335 4sqlem10 12339 lgsvalmod 13714 lgsdir2lem1 13723 lgsdir2lem4 13726 lgsdir2lem5 13727 lgsdirprm 13729 lgsne0 13733 |
Copyright terms: Public domain | W3C validator |