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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8790 | . . 3 | |
2 | 1re 7765 | . . 3 | |
3 | 2pos 8811 | . . 3 | |
4 | 0lt1 7889 | . . 3 | |
5 | 1, 2, 3, 4 | addgt0ii 8253 | . 2 |
6 | df-3 8780 | . 2 | |
7 | 5, 6 | breqtrri 3955 | 1 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3929 (class class class)co 5774 cc0 7620 c1 7621 caddc 7623 clt 7800 c2 8771 c3 8772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-2 8779 df-3 8780 |
This theorem is referenced by: 3ne0 8815 3ap0 8816 4pos 8817 8th4div3 8939 halfpm6th 8940 3rp 9447 sqrt9 10820 ef01bndlem 11463 cos2bnd 11467 sin01gt0 11468 cos01gt0 11469 flodddiv4 11631 coseq0negpitopi 12917 tangtx 12919 sincos6thpi 12923 cos02pilt1 12932 ex-gcd 12943 |
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