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| Mirrors > Home > MPE Home > Th. List > nn0rei | Structured version Visualization version GIF version | ||
| Description: A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.) |
| Ref | Expression |
|---|---|
| nn0rei.1 | ⊢ 𝐴 ∈ ℕ0 |
| Ref | Expression |
|---|---|
| nn0rei | ⊢ 𝐴 ∈ ℝ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0ssre 12530 | . 2 ⊢ ℕ0 ⊆ ℝ | |
| 2 | nn0rei.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
| 3 | 1, 2 | sselii 3980 | 1 ⊢ 𝐴 ∈ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ℝcr 11154 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-n0 12527 |
| This theorem is referenced by: nn0lele2xi 12582 numlt 12758 numltc 12759 decle 12767 decleh 12768 nn0le2msqi 14306 nn0opthlem2 14308 nn0opthi 14309 faclbnd4lem1 14332 hashunlei 14464 hashsslei 14465 fsumcube 16096 divalglem5 16434 prmreclem3 16956 prmreclem5 16958 modxai 17106 modsubi 17110 prmlem2 17157 slotsbhcdif 17459 slotsbhcdifOLD 17460 cnfldfunALTOLDOLD 21393 psdmul 22170 dscmet 24585 tnglemOLD 24654 log2ublem1 26989 log2ub 26992 log2le1 26993 birthday 26997 ppiublem1 27246 ppiub 27248 bpos1lem 27326 bpos1 27327 bpos 27337 vdegp1bi 29555 9p10ne21 30489 dp20u 32860 rpdp2cl 32864 dp2lt10 32866 dp2lt 32867 dp2ltsuc 32868 dp2ltc 32869 dpmul100 32879 dp3mul10 32880 dpmul1000 32881 dpgti 32888 dpadd2 32892 dpadd 32893 dpadd3 32894 dpmul 32895 dpmul4 32896 hgt750lemd 34663 hgt750lem 34666 hgt750leme 34673 tgoldbachgnn 34674 resqrtvalex 43658 imsqrtvalex 43659 fmtno4prmfac 47559 31prm 47584 evengpoap3 47786 ackval42 48617 prstcocvalOLD 49161 |
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