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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 12103 | . 2 ⊢ (9 + 1) = ;10 | |
2 | 9nn 11738 | . . 3 ⊢ 9 ∈ ℕ | |
3 | peano2nn 11652 | . . 3 ⊢ (9 ∈ ℕ → (9 + 1) ∈ ℕ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (9 + 1) ∈ ℕ |
5 | 1, 4 | eqeltrri 2912 | 1 ⊢ ;10 ∈ ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 ℕcn 11640 9c9 11702 ;cdc 12101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-dec 12102 |
This theorem is referenced by: 10pos 12118 decnncl2 12125 declt 12129 decltc 12130 declti 12139 dec10p 12144 3dec 13629 3dvds 15682 163prm 16460 631prm 16462 plendx 16668 pleid 16669 otpsstr 16670 ressle 16674 odrngstr 16681 imasvalstr 16727 isposix 17569 ipostr 17765 cnfldstr 20549 bclbnd 25858 ex-prmo 28240 rpdp2cl 30560 dp2ltsuc 30564 dpmul10 30573 decdiv10 30574 dpmul100 30575 dp3mul10 30576 dpadd2 30588 dpadd 30589 dpadd3 30590 dpmul 30591 dpmul4 30592 oppgle 30642 hgt750lem 31924 tgoldbachgt 31936 rmydioph 39618 1t10e1p1e11 43517 257prm 43730 127prm 43770 3exp4mod41 43788 41prothprmlem1 43789 bgoldbtbndlem1 43977 bgoldbachlt 43985 tgblthelfgott 43987 tgoldbachlt 43988 |
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