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Mirrors > Home > ILE Home > Th. List > 3pos | Unicode version |
Description: The number 3 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
3pos |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8987 |
. . 3
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2 | 1re 7955 |
. . 3
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3 | 2pos 9008 |
. . 3
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4 | 0lt1 8082 |
. . 3
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5 | 1, 2, 3, 4 | addgt0ii 8446 |
. 2
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6 | df-3 8977 |
. 2
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7 | 5, 6 | breqtrri 4030 |
1
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Colors of variables: wff set class |
Syntax hints: class class
class wbr 4003 (class class class)co 5874
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This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-iota 5178 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-2 8976 df-3 8977 |
This theorem is referenced by: 3ne0 9012 3ap0 9013 4pos 9014 8th4div3 9136 halfpm6th 9137 3rp 9657 fz0to4untppr 10121 sqrt9 11052 ef01bndlem 11759 cos2bnd 11763 sin01gt0 11764 cos01gt0 11765 flodddiv4 11933 slotsdifunifndx 12677 coseq0negpitopi 14150 tangtx 14152 sincos6thpi 14156 cos02pilt1 14165 lgsdir2lem1 14322 ex-gcd 14365 |
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