![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12534 | . 2 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 12533 | . 2 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12738 | 1 ⊢ ;10 ∈ ℕ0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 0cc0 11149 1c1 11150 ℕ0cn0 12518 ;cdc 12723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-ltxr 11294 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-dec 12724 |
This theorem is referenced by: decnncl 12743 dec0u 12744 dec0h 12745 decsuc 12754 decle 12757 decma 12774 decmac 12775 decma2c 12776 decadd 12777 decaddc 12778 decsubi 12786 decmul1c 12788 decmul2c 12789 decmul10add 12792 9t11e99 12853 sq10 14276 dec2dvds 17060 decsplit0b 17077 decsplit1 17079 decsplit 17080 karatsuba 17081 139prm 17121 317prm 17123 1259lem1 17128 1259lem3 17130 2503lem1 17134 4001lem1 17138 4001lem3 17140 9p10ne21 30400 dfdec100 32734 dp20u 32742 dp20h 32743 dp2clq 32745 dpmul100 32761 dpmul1000 32763 dpexpp1 32772 0dp2dp 32773 dpmul 32777 dpmul4 32778 hgt750lemd 34507 hgt750lem2 34511 hgt750leme 34517 tgoldbachgnn 34518 aks4d1p1p7 41786 sqdeccom12 42028 rmydioph 42709 tgoldbach 47425 |
Copyright terms: Public domain | W3C validator |