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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 8981 | . 2 ⊢ ℕ0 ⊆ ℝ | |
2 | nn0red.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
3 | 1, 2 | sseldi 3095 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ℝcr 7619 ℕ0cn0 8977 |
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 2121 ax-sep 4046 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-int 3772 df-inn 8721 df-n0 8978 |
This theorem is referenced by: nn0cnd 9032 nn0readdcl 9036 nn01to3 9409 flqmulnn0 10072 modifeq2int 10159 modaddmodup 10160 modaddmodlo 10161 modsumfzodifsn 10169 expnegap0 10301 nn0le2msqd 10465 nn0opthlem2d 10467 nn0opthd 10468 faclbnd6 10490 bcval5 10509 filtinf 10538 zfz1isolemiso 10582 mertenslemi1 11304 efcllemp 11364 eftlub 11396 oddge22np1 11578 nn0oddm1d2 11606 gcdaddm 11672 bezoutlemsup 11697 gcdzeq 11710 dvdssqlem 11718 nn0seqcvgd 11722 lcmneg 11755 mulgcddvds 11775 qredeu 11778 pw2dvdseulemle 11845 pw2dvdseu 11846 nn0sqrtelqelz 11884 nonsq 11885 ennnfoneleminc 11924 ennnfonelemkh 11925 ennnfonelemex 11927 ennnfonelemim 11937 |
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