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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 12333 | . 2 ⊢ 6 = (5 + 1) | |
| 2 | 5nn 12352 | . . 3 ⊢ 5 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (5 ∈ ℕ → (5 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (5 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltri 2837 | 1 ⊢ 6 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 5c5 12324 6c6 12325 |
| 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 |
| 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-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 |
| This theorem is referenced by: 7nn 12358 6nn0 12547 ef01bndlem 16220 sin01bnd 16221 cos01bnd 16222 6gcd4e2 16575 6lcm4e12 16653 83prm 17160 139prm 17161 163prm 17162 prmo6 17167 vscandx 17363 vscaid 17364 lmodstr 17369 ipsstr 17380 lt6abl 19913 psrvalstr 21936 opsrvscaOLD 22076 tngvscaOLD 24665 sincos3rdpi 26559 1cubrlem 26884 quart1cl 26897 quart1lem 26898 quart1 26899 log2ub 26992 log2le1 26993 basellem5 27128 basellem8 27131 basellem9 27132 ppiublem1 27246 ppiublem2 27247 ppiub 27248 bpos1 27327 bposlem9 27336 itvndx 28445 itvid 28447 slotsinbpsd 28449 lngndxnitvndx 28451 trkgstr 28452 ttgvalOLD 28884 ttglemOLD 28886 ttgvscaOLD 28893 ttgdsOLD 28895 eengstr 28995 ex-cnv 30456 ex-dm 30458 ex-dvds 30475 ex-gcd 30476 ex-lcm 30477 resvvscaOLD 33364 hgt750lem 34666 60gcd6e6 42005 60gcd7e1 42006 12lcm5e60 42009 60lcm6e60 42010 60lcm7e420 42011 lcm6un 42019 lcmineqlem 42053 3lexlogpow5ineq1 42055 aks4d1p1p5 42076 aks4d1p1 42077 rmydioph 43026 expdiophlem2 43034 algstr 43185 mnringvscadOLD 44244 139prmALT 47583 31prm 47584 127prm 47586 6even 47698 gbowge7 47750 stgoldbwt 47763 sbgoldbwt 47764 mogoldbb 47772 sbgoldbo 47774 nnsum3primesle9 47781 nnsum4primeseven 47787 wtgoldbnnsum4prm 47789 bgoldbnnsum3prm 47791 zlmodzxzequa 48413 zlmodzxznm 48414 zlmodzxzequap 48416 zlmodzxzldeplem3 48419 zlmodzxzldep 48421 ldepsnlinclem2 48423 ldepsnlinc 48425 |
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