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Mirrors > Home > ILE Home > Th. List > nn0red | GIF version |
Description: A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0red | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssre 9109 | . 2 ⊢ ℕ0 ⊆ ℝ | |
2 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sseldi 3135 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 ℝcr 7743 ℕ0cn0 9105 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-rnegex 7853 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-int 3819 df-inn 8849 df-n0 9106 |
This theorem is referenced by: nn0cnd 9160 nn0readdcl 9164 nn01to3 9546 xnn0dcle 9729 flqmulnn0 10224 modifeq2int 10311 modaddmodup 10312 modaddmodlo 10313 modsumfzodifsn 10321 expnegap0 10453 nn0leexp2 10613 nn0le2msqd 10621 nn0opthlem2d 10623 nn0opthd 10624 faclbnd6 10646 bcval5 10665 filtinf 10694 zfz1isolemiso 10738 mertenslemi1 11462 efcllemp 11585 eftlub 11617 oddge22np1 11803 nn0oddm1d2 11831 gcdaddm 11902 bezoutlemsup 11927 gcdzeq 11940 dvdssqlem 11948 nn0seqcvgd 11952 lcmneg 11985 mulgcddvds 12005 qredeu 12008 pw2dvdseulemle 12076 pw2dvdseu 12077 nn0sqrtelqelz 12115 nonsq 12116 pythagtriplem3 12176 pythagtriplem6 12179 pythagtriplem7 12180 pclemub 12196 pcprendvds 12199 pcpremul 12202 pcidlem 12231 pcgcd1 12236 pc2dvds 12238 pcz 12240 pcprmpw2 12241 fldivp1 12255 pcfaclem 12256 pcfac 12257 pcbc 12258 ennnfoneleminc 12281 ennnfonelemkh 12282 ennnfonelemex 12284 ennnfonelemim 12294 |
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