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Mirrors > Home > MPE Home > Th. List > 5nn | Structured version Visualization version GIF version |
Description: 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
5nn | ⊢ 5 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 11692 | . 2 ⊢ 5 = (4 + 1) | |
2 | 4nn 11709 | . . 3 ⊢ 4 ∈ ℕ | |
3 | peano2nn 11639 | . . 3 ⊢ (4 ∈ ℕ → (4 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (4 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2909 | 1 ⊢ 5 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10527 + caddc 10529 ℕcn 11627 4c4 11683 5c5 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 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 |
This theorem is referenced by: 6nn 11715 5nn0 11906 prm23ge5 16142 dec5dvds 16390 dec5nprm 16392 dec2nprm 16393 5prm 16432 10nprm 16437 23prm 16442 prmlem2 16443 43prm 16445 83prm 16446 317prm 16449 prmo5 16452 scandx 16622 scaid 16623 lmodstr 16626 ipsstr 16633 resssca 16640 ccondx 16679 ccoid 16680 ressco 16682 slotsbhcdif 16683 prdsvalstr 16716 oppchomfval 16974 oppcbas 16978 rescco 17092 catstr 17217 lt6abl 18946 mgpsca 19177 psrvalstr 20073 opsrsca 20193 tngsca 23183 log2ublem1 25452 log2ublem2 25453 log2ub 25455 birthday 25460 ppiublem1 25706 ppiublem2 25707 ppiub 25708 bclbnd 25784 bposlem3 25790 bposlem4 25791 bposlem5 25792 bposlem6 25793 bposlem8 25795 bposlem9 25796 lgsdir2lem3 25831 ex-eprel 28140 ex-xp 28143 fib6 31564 hgt750lem2 31823 hgt750leme 31829 rmydioph 39491 expdiophlem2 39499 algstr 39657 inductionexd 40385 257prm 43570 fmtno4prmfac193 43582 31prm 43607 41prothprm 43631 gbowge7 43775 gbege6 43777 stgoldbwt 43788 sbgoldbwt 43789 sbgoldbm 43796 sbgoldbo 43799 nnsum3primesle9 43806 |
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