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Mirrors > Home > MPE Home > Th. List > 6nn | Structured version Visualization version GIF version |
Description: 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
6nn | ⊢ 6 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-6 11380 | . 2 ⊢ 6 = (5 + 1) | |
2 | 5nn 11401 | . . 3 ⊢ 5 ∈ ℕ | |
3 | peano2nn 11326 | . . 3 ⊢ (5 ∈ ℕ → (5 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (5 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2874 | 1 ⊢ 6 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 (class class class)co 6878 1c1 10225 + caddc 10227 ℕcn 11312 5c5 11371 6c6 11372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-1cn 10282 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 |
This theorem is referenced by: 7nn 11409 6nn0 11603 ef01bndlem 15250 sin01bnd 15251 cos01bnd 15252 6gcd4e2 15590 6lcm4e12 15664 83prm 16157 139prm 16158 163prm 16159 prmo6 16164 vscandx 16336 vscaid 16337 lmodstr 16338 ipsstr 16345 ressvsca 16353 lt6abl 18611 psrvalstr 19686 opsrvsca 19804 tngvsca 22778 sincos3rdpi 24610 1cubrlem 24920 quart1cl 24933 quart1lem 24934 quart1 24935 log2ub 25028 log2le1 25029 basellem5 25163 basellem8 25166 basellem9 25167 ppiublem1 25279 ppiublem2 25280 ppiub 25281 bpos1 25360 bposlem9 25369 itvndx 25691 itvid 25693 trkgstr 25695 ttgval 26112 ttglem 26113 ttgvsca 26117 ttgds 26118 eengstr 26217 ex-cnv 27822 ex-dm 27824 ex-dvds 27841 ex-gcd 27842 ex-lcm 27843 resvvsca 30350 hgt750lem 31249 rmydioph 38366 expdiophlem2 38374 algstr 38532 139prmALT 42293 31prm 42294 127prm 42297 6even 42402 gbowge7 42433 stgoldbwt 42446 sbgoldbwt 42447 mogoldbb 42455 sbgoldbo 42457 nnsum3primesle9 42464 nnsum4primeseven 42470 wtgoldbnnsum4prm 42472 bgoldbnnsum3prm 42474 zlmodzxzequa 43084 zlmodzxznm 43085 zlmodzxzequap 43087 zlmodzxzldeplem3 43090 zlmodzxzldep 43092 ldepsnlinclem2 43094 ldepsnlinc 43096 |
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