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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 12471 | . 2 ⊢ ℕ0 ⊆ ℝ | |
2 | nn0rei.1 | . 2 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | sselii 3977 | 1 ⊢ 𝐴 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ℝcr 11104 ℕ0cn0 12467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-i2m1 11173 ax-1ne0 11174 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-nn 12208 df-n0 12468 |
This theorem is referenced by: nn0le2xi 12521 nn0lele2xi 12522 numlt 12697 numltc 12698 decle 12706 decleh 12707 nn0le2msqi 14222 nn0opthlem2 14224 nn0opthi 14225 faclbnd4lem1 14248 hashunlei 14380 hashsslei 14381 fsumcube 15999 divalglem5 16335 prmreclem3 16846 prmreclem5 16848 modxai 16996 modsubi 17000 prmlem2 17048 slotsbhcdif 17355 slotsbhcdifOLD 17356 cnfldfunALTOLD 20942 dscmet 24062 tnglemOLD 24131 log2ublem1 26430 log2ub 26433 log2le1 26434 birthday 26438 ppiublem1 26684 ppiub 26686 bpos1lem 26764 bpos1 26765 bpos 26775 vdegp1bi 28773 9p10ne21 29702 dp20u 32021 rpdp2cl 32025 dp2lt10 32027 dp2lt 32028 dp2ltsuc 32029 dp2ltc 32030 dpmul100 32040 dp3mul10 32041 dpmul1000 32042 dpgti 32049 dpadd2 32053 dpadd 32054 dpadd3 32055 dpmul 32056 dpmul4 32057 hgt750lemd 33597 hgt750lem 33600 hgt750leme 33607 tgoldbachgnn 33608 resqrtvalex 42328 imsqrtvalex 42329 fmtno4prmfac 46174 31prm 46199 evengpoap3 46401 ackval42 47283 prstcocvalOLD 47593 |
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