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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 11691 | . 2 ⊢ 5 = (4 + 1) | |
2 | 4nn 11708 | . . 3 ⊢ 4 ∈ ℕ | |
3 | peano2nn 11638 | . . 3 ⊢ (4 ∈ ℕ → (4 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (4 + 1) ∈ ℕ |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 5 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 (class class class)co 7145 1c1 10526 + caddc 10528 ℕcn 11626 4c4 11682 5c5 11683 |
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 |
This theorem is referenced by: 6nn 11714 5nn0 11905 prm23ge5 16140 dec5dvds 16388 dec5nprm 16390 dec2nprm 16391 5prm 16430 10nprm 16435 23prm 16440 prmlem2 16441 43prm 16443 83prm 16444 317prm 16447 prmo5 16450 scandx 16620 scaid 16621 lmodstr 16624 ipsstr 16631 resssca 16638 ccondx 16677 ccoid 16678 ressco 16680 slotsbhcdif 16681 prdsvalstr 16714 oppchomfval 16972 oppcbas 16976 rescco 17090 catstr 17215 lt6abl 18944 mgpsca 19175 psrvalstr 20071 opsrsca 20191 tngsca 23181 log2ublem1 25451 log2ublem2 25452 log2ub 25454 birthday 25459 ppiublem1 25705 ppiublem2 25706 ppiub 25707 bclbnd 25783 bposlem3 25789 bposlem4 25790 bposlem5 25791 bposlem6 25792 bposlem8 25794 bposlem9 25795 lgsdir2lem3 25830 ex-eprel 28139 ex-xp 28142 fib6 31563 hgt750lem2 31822 hgt750leme 31828 rmydioph 39489 expdiophlem2 39497 algstr 39655 inductionexd 40383 257prm 43600 fmtno4prmfac193 43612 31prm 43637 41prothprm 43661 gbowge7 43805 gbege6 43807 stgoldbwt 43818 sbgoldbwt 43819 sbgoldbm 43826 sbgoldbo 43829 nnsum3primesle9 43836 |
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