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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 11692 | . 2 ⊢ 6 = (5 + 1) | |
2 | 5nn 11711 | . . 3 ⊢ 5 ∈ ℕ | |
3 | peano2nn 11638 | . . 3 ⊢ (5 ∈ ℕ → (5 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (5 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 6 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10526 + caddc 10528 ℕcn 11626 5c5 11683 6c6 11684 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 |
This theorem is referenced by: 7nn 11717 6nn0 11906 ef01bndlem 15525 sin01bnd 15526 cos01bnd 15527 6gcd4e2 15874 6lcm4e12 15948 83prm 16444 139prm 16445 163prm 16446 prmo6 16451 vscandx 16622 vscaid 16623 lmodstr 16624 ipsstr 16631 ressvsca 16639 lt6abl 18944 psrvalstr 20071 opsrvsca 20190 tngvsca 23182 sincos3rdpi 25029 1cubrlem 25346 quart1cl 25359 quart1lem 25360 quart1 25361 log2ub 25454 log2le1 25455 basellem5 25589 basellem8 25592 basellem9 25593 ppiublem1 25705 ppiublem2 25706 ppiub 25707 bpos1 25786 bposlem9 25795 itvndx 26153 itvid 26155 trkgstr 26157 ttgval 26588 ttglem 26589 ttgvsca 26593 ttgds 26594 eengstr 26693 ex-cnv 28143 ex-dm 28145 ex-dvds 28162 ex-gcd 28163 ex-lcm 28164 resvvsca 30834 hgt750lem 31821 rmydioph 39489 expdiophlem2 39497 algstr 39655 139prmALT 43636 31prm 43637 127prm 43640 6even 43753 gbowge7 43805 stgoldbwt 43818 sbgoldbwt 43819 mogoldbb 43827 sbgoldbo 43829 nnsum3primesle9 43836 nnsum4primeseven 43842 wtgoldbnnsum4prm 43844 bgoldbnnsum3prm 43846 zlmodzxzequa 44479 zlmodzxznm 44480 zlmodzxzequap 44482 zlmodzxzldeplem3 44485 zlmodzxzldep 44487 ldepsnlinclem2 44489 ldepsnlinc 44491 |
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