Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11693 | . 2 ⊢ 6 = (5 + 1) | |
2 | 5nn 11712 | . . 3 ⊢ 5 ∈ ℕ | |
3 | peano2nn 11639 | . . 3 ⊢ (5 ∈ ℕ → (5 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (5 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2909 | 1 ⊢ 6 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10527 + caddc 10529 ℕcn 11627 5c5 11684 6c6 11685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-1cn 10584 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 |
This theorem is referenced by: 7nn 11718 6nn0 11907 ef01bndlem 15527 sin01bnd 15528 cos01bnd 15529 6gcd4e2 15876 6lcm4e12 15950 83prm 16446 139prm 16447 163prm 16448 prmo6 16453 vscandx 16624 vscaid 16625 lmodstr 16626 ipsstr 16633 ressvsca 16641 lt6abl 18946 psrvalstr 20073 opsrvsca 20192 tngvsca 23184 sincos3rdpi 25031 1cubrlem 25346 quart1cl 25359 quart1lem 25360 quart1 25361 log2ub 25455 log2le1 25456 basellem5 25590 basellem8 25593 basellem9 25594 ppiublem1 25706 ppiublem2 25707 ppiub 25708 bpos1 25787 bposlem9 25796 itvndx 26154 itvid 26156 trkgstr 26158 ttgval 26589 ttglem 26590 ttgvsca 26594 ttgds 26595 eengstr 26694 ex-cnv 28144 ex-dm 28146 ex-dvds 28163 ex-gcd 28164 ex-lcm 28165 resvvsca 30835 hgt750lem 31822 rmydioph 39491 expdiophlem2 39499 algstr 39657 139prmALT 43606 31prm 43607 127prm 43610 6even 43723 gbowge7 43775 stgoldbwt 43788 sbgoldbwt 43789 mogoldbb 43797 sbgoldbo 43799 nnsum3primesle9 43806 nnsum4primeseven 43812 wtgoldbnnsum4prm 43814 bgoldbnnsum3prm 43816 zlmodzxzequa 44449 zlmodzxznm 44450 zlmodzxzequap 44452 zlmodzxzldeplem3 44455 zlmodzxzldep 44457 ldepsnlinclem2 44459 ldepsnlinc 44461 |
Copyright terms: Public domain | W3C validator |