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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 11823 | . 2 ⊢ (9 + 1) = ;10 | |
2 | 9nn 11455 | . . 3 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11364 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (9 + 1) ∈ ℕ |
5 | 1, 4 | eqeltrri 2903 | 1 ⊢ ;10 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2166 (class class class)co 6905 0cc0 10252 1c1 10253 + caddc 10255 ℕcn 11350 9c9 11413 ;cdc 11821 |
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 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 |
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 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-ltxr 10396 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-dec 11822 |
This theorem is referenced by: 10pos 11838 10reOLD 11841 decnncl2 11846 declt 11850 decltc 11851 declti 11860 dec10p 11865 3dvds 15429 163prm 16197 631prm 16199 plendx 16406 pleid 16407 otpsstr 16408 ressle 16412 odrngstr 16419 imasvalstr 16465 isposix 17310 ipostr 17506 cnfldstr 20108 bclbnd 25418 ex-prmo 27874 rpdp2cl 30135 dp2ltsuc 30139 dpmul10 30148 decdiv10 30149 dpmul100 30150 dp3mul10 30151 dpadd2 30163 dpadd 30164 dpadd3 30165 dpmul 30166 dpmul4 30167 oppgle 30198 hgt750lem 31278 tgoldbachgt 31290 rmydioph 38424 1t10e1p1e11 42208 257prm 42303 127prm 42345 3exp4mod41 42363 41prothprmlem1 42364 bgoldbtbndlem1 42523 bgoldbachlt 42531 tgblthelfgott 42533 tgoldbachlt 42534 |
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