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Mirrors > Home > MPE Home > Th. List > nnge1 | Structured version Unicode version |
Description: A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.) |
Ref | Expression |
---|---|
nnge1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4394 |
. 2
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2 | breq2 4394 |
. 2
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3 | breq2 4394 |
. 2
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4 | breq2 4394 |
. 2
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5 | 1le1 10065 |
. 2
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6 | nnre 10430 |
. . 3
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7 | recn 9473 |
. . . . . 6
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8 | 7 | addid1d 9670 |
. . . . 5
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9 | 8 | breq2d 4402 |
. . . 4
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10 | 0lt1 9963 |
. . . . . . . 8
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11 | 0re 9487 |
. . . . . . . . 9
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12 | 1re 9486 |
. . . . . . . . 9
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13 | axltadd 9549 |
. . . . . . . . 9
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14 | 11, 12, 13 | mp3an12 1305 |
. . . . . . . 8
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15 | 10, 14 | mpi 17 |
. . . . . . 7
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16 | readdcl 9466 |
. . . . . . . . 9
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17 | 11, 16 | mpan2 671 |
. . . . . . . 8
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18 | peano2re 9643 |
. . . . . . . 8
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19 | lttr 9552 |
. . . . . . . . 9
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20 | 12, 19 | mp3an3 1304 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 17, 18, 20 | syl2anc 661 |
. . . . . . 7
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22 | 15, 21 | mpand 675 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | con3d 133 |
. . . . 5
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24 | lenlt 9554 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 12, 17, 24 | sylancr 663 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | lenlt 9554 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 12, 18, 26 | sylancr 663 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 23, 25, 27 | 3imtr4d 268 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 9, 28 | sylbird 235 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 6, 29 | syl 16 |
. 2
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31 | 1, 2, 3, 4, 5, 30 | nnind 10441 |
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 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 ax-resscn 9440 ax-1cn 9441 ax-icn 9442 ax-addcl 9443 ax-addrcl 9444 ax-mulcl 9445 ax-mulrcl 9446 ax-mulcom 9447 ax-addass 9448 ax-mulass 9449 ax-distr 9450 ax-i2m1 9451 ax-1ne0 9452 ax-1rid 9453 ax-rnegex 9454 ax-rrecex 9455 ax-cnre 9456 ax-pre-lttri 9457 ax-pre-lttrn 9458 ax-pre-ltadd 9459 ax-pre-mulgt0 9460 |
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 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-pss 3442 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-tp 3980 df-op 3982 df-uni 4190 df-iun 4271 df-br 4391 df-opab 4449 df-mpt 4450 df-tr 4484 df-eprel 4730 df-id 4734 df-po 4739 df-so 4740 df-fr 4777 df-we 4779 df-ord 4820 df-on 4821 df-lim 4822 df-suc 4823 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-riota 6151 df-ov 6193 df-oprab 6194 df-mpt2 6195 df-om 6577 df-recs 6932 df-rdg 6966 df-er 7201 df-en 7411 df-dom 7412 df-sdom 7413 df-pnf 9521 df-mnf 9522 df-xr 9523 df-ltxr 9524 df-le 9525 df-sub 9698 df-neg 9699 df-nn 10424 |
This theorem is referenced by: nngt1ne1 10450 nnle1eq1 10451 nngt0 10452 nnnlt1 10453 nnrecgt0 10460 nnge1d 10465 elnnnn0c 10726 elnnz1 10773 zltp1le 10795 elfz1b 11628 fzo1fzo0n0 11689 fzo0sn0fzo1 11718 nnlesq 12070 digit1 12099 faclbnd 12167 faclbnd3 12169 faclbnd4lem1 12170 faclbnd4lem4 12173 fstwrdne0 12366 swrdn0 12426 swrdtrcfv 12439 swrdccatwrd 12464 divalglem1 13700 isprm3 13874 pockthg 14069 infpn2 14076 dscmet 20281 ovolunlem1a 21095 vitali 21209 plyeq0lem 21794 logtayllem 22220 leibpi 22453 vmalelog 22660 chtublem 22666 logfaclbnd 22677 bposlem1 22739 dchrisum0lem1 22881 logdivbnd 22921 pntlemn 22965 ostth2lem3 23000 nexple 26582 eulerpartlems 26877 eulerpartlemb 26885 ballotlem2 27005 plymulx0 27082 fz0n 27523 nndivlub 28438 fzsplit1nn0 29230 pell1qrgaplem 29352 pellqrex 29358 monotoddzzfi 29421 jm2.23 29483 wallispilem4 30001 wallispilem5 30002 wallispi 30003 wallispi2lem1 30004 stirlinglem5 30011 stirlinglem13 30019 clwwisshclwwlem 30608 clwlkfclwwlk 30655 chfacfpmmulgsum2 31319 |
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