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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 12735 | . 2 ⊢ (9 + 1) = ;10 | |
| 2 | 9nn 12364 | . . 3 ⊢ 9 ∈ ℕ | |
| 3 | peano2nn 12278 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (9 + 1) ∈ ℕ |
| 5 | 1, 4 | eqeltrri 2838 | 1 ⊢ ;10 ∈ ℕ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ℕcn 12266 9c9 12328 ;cdc 12733 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-dec 12734 |
| This theorem is referenced by: 10pos 12750 decnncl2 12757 declt 12761 decltc 12762 declti 12771 dec10p 12776 3dec 14305 3dvds 16368 163prm 17162 631prm 17164 plendx 17410 pleid 17411 plendxnn 17412 otpsstr 17420 odrngstr 17447 imasvalstr 17496 isposixOLD 18371 ipostr 18574 cnfldstr 21366 cnfldstrOLD 21381 bclbnd 27324 ex-prmo 30478 rpdp2cl 32864 dp2ltsuc 32868 dpmul10 32877 decdiv10 32878 dpmul100 32879 dp3mul10 32880 dpadd2 32892 dpadd 32893 dpadd3 32894 dpmul 32895 dpmul4 32896 oppgleOLD 32952 idlsrgstr 33530 hgt750lem 34666 tgoldbachgt 34678 60gcd6e6 42005 aks4d1p1p7 42075 rmydioph 43026 1t10e1p1e11 47322 257prm 47548 127prm 47586 3exp4mod41 47603 41prothprmlem1 47604 bgoldbtbndlem1 47792 bgoldbachlt 47800 tgblthelfgott 47802 tgoldbachlt 47803 |
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