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Mirrors > Home > MPE Home > Th. List > 10nn0 | Structured version Visualization version GIF version |
Description: 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
10nn0 | ⊢ ;10 ∈ ℕ0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 12539 | . 2 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 12538 | . 2 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12745 | 1 ⊢ ;10 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 0cc0 11152 1c1 11153 ℕ0cn0 12523 ;cdc 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 |
This theorem is referenced by: decnncl 12750 dec0u 12751 dec0h 12752 decsuc 12761 decle 12764 decma 12781 decmac 12782 decma2c 12783 decadd 12784 decaddc 12785 decsubi 12793 decmul1c 12795 decmul2c 12796 decmul10add 12799 9t11e99 12860 sq10 14299 dec2dvds 17096 decsplit0b 17113 decsplit1 17115 decsplit 17116 karatsuba 17117 139prm 17157 317prm 17159 1259lem1 17164 1259lem3 17166 2503lem1 17170 4001lem1 17174 4001lem3 17176 9p10ne21 30498 dfdec100 32836 dp20u 32844 dp20h 32845 dp2clq 32847 dpmul100 32863 dpmul1000 32865 dpexpp1 32874 0dp2dp 32875 dpmul 32879 dpmul4 32880 hgt750lemd 34641 hgt750lem2 34645 hgt750leme 34651 tgoldbachgnn 34652 aks4d1p1p7 42055 sqdeccom12 42302 rmydioph 43002 tgoldbach 47741 gpg5grlic 47974 |
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