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Mirrors > Home > MPE Home > Th. List > nnrei | Structured version Visualization version GIF version |
Description: A positive integer is a real number. (Contributed by NM, 18-Aug-1999.) |
Ref | Expression |
---|---|
nnre.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
nnrei | ⊢ 𝐴 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre.1 | . 2 ⊢ 𝐴 ∈ ℕ | |
2 | nnre 11359 | . 2 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 ℝcr 10252 ℕcn 11351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-i2m1 10321 ax-1ne0 10322 ax-rrecex 10325 ax-cnre 10326 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-nn 11352 |
This theorem is referenced by: nncniOLD 11363 nnne0i 11392 10reOLD 11842 numlt 11848 numltc 11849 faclbnd4lem1 13374 ef01bndlem 15287 dvdslelem 15409 divalglem6 15496 pockthi 15983 modsubi 16148 prmlem1 16181 prmlem2 16193 strleun 16332 strle1 16333 oppchomfval 16727 oppcbas 16731 rescco 16845 opprlem 18983 sralem 19539 opsrbaslem 19839 zlmlem 20226 znbaslem 20247 tnglem 22815 log2ublem1 25087 log2ublem2 25088 log2ub 25090 bpos1lem 25421 bposlem8 25430 bposlem9 25431 ttgval 26175 ttglem 26176 cchhllem 26187 slotsbaseefdif 26294 structvtxvallem 26319 lmat22e12 30431 lmat22e21 30432 lmat22e22 30433 ballotlem2 31097 ballotlem5 31108 ballotth 31146 chtvalz 31257 hgt750lem 31279 tgoldbachgt 31291 cnndvlem1 33061 hlhilslem 38014 jm2.27dlem2 38421 bgoldbtbndlem1 42524 tgblthelfgott 42534 tgoldbachlt 42535 |
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