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Mirrors > Home > MPE Home > Th. List > nn0zi | Structured version Unicode version |
Description: A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
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nn0zi.1 |
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Ref | Expression |
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nn0zi |
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Step | Hyp | Ref | Expression |
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1 | nn0ssz 10770 |
. 2
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2 | nn0zi.1 |
. 2
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3 | 1, 2 | sselii 3453 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-pss 3444 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-tp 3982 df-op 3984 df-uni 4192 df-iun 4273 df-br 4393 df-opab 4451 df-mpt 4452 df-tr 4486 df-eprel 4732 df-id 4736 df-po 4741 df-so 4742 df-fr 4779 df-we 4781 df-ord 4822 df-on 4823 df-lim 4824 df-suc 4825 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-om 6579 df-recs 6934 df-rdg 6968 df-er 7203 df-en 7413 df-dom 7414 df-sdom 7415 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-nn 10426 df-n0 10683 df-z 10750 |
This theorem is referenced by: fzo0to42pr 11719 expnass 12074 faclbnd4lem1 12172 efsep 13498 divalglem0 13701 divalglem2 13703 ndvdsi 13718 gcdaddmlem 13816 phicl2 13947 dec2dvds 14196 dec5dvds2 14198 modxai 14201 mod2xnegi 14204 gcdi 14206 gcdmodi 14207 1259lem1 14259 1259lem2 14260 1259lem3 14261 1259lem4 14262 1259lem5 14263 2503lem1 14265 2503lem2 14266 2503lem3 14267 4001lem1 14269 4001lem2 14270 4001lem3 14271 4001lem4 14272 strlemor1 14369 ppi1i 22624 ppi2i 22625 ppiublem1 22659 4cycl4v4e 23689 4cycl4dv4e 23691 ballotlemfelz 27009 wallispi2lem1 30006 linevalexample 31004 |
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