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Mirrors > Home > MPE Home > Th. List > 10nn | Structured version Visualization version GIF version |
Description: 10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
10nn | ⊢ ;10 ∈ ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9p1e10 11824 | . 2 ⊢ (9 + 1) = ;10 | |
2 | 9nn 11456 | . . 3 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11365 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (9 + 1) ∈ ℕ |
5 | 1, 4 | eqeltrri 2904 | 1 ⊢ ;10 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 (class class class)co 6906 0cc0 10253 1c1 10254 + caddc 10256 ℕcn 11351 9c9 11414 ;cdc 11822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-ltxr 10397 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-dec 11823 |
This theorem is referenced by: 10pos 11839 10reOLD 11842 decnncl2 11847 declt 11851 decltc 11852 declti 11861 dec10p 11866 3dvds 15430 163prm 16198 631prm 16200 plendx 16407 pleid 16408 otpsstr 16409 ressle 16413 odrngstr 16420 imasvalstr 16466 isposix 17311 ipostr 17507 cnfldstr 20109 bclbnd 25419 ex-prmo 27875 rpdp2cl 30136 dp2ltsuc 30140 dpmul10 30149 decdiv10 30150 dpmul100 30151 dp3mul10 30152 dpadd2 30164 dpadd 30165 dpadd3 30166 dpmul 30167 dpmul4 30168 oppgle 30199 hgt750lem 31279 tgoldbachgt 31291 rmydioph 38425 1t10e1p1e11 42209 257prm 42304 127prm 42346 3exp4mod41 42364 41prothprmlem1 42365 bgoldbtbndlem1 42524 bgoldbachlt 42532 tgblthelfgott 42534 tgoldbachlt 42535 |
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