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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 8783 | . . 3 | |
2 | 1re 7758 | . . 3 | |
3 | 2pos 8804 | . . 3 | |
4 | 0lt1 7882 | . . 3 | |
5 | 1, 2, 3, 4 | addgt0ii 8246 | . 2 |
6 | df-3 8773 | . 2 | |
7 | 5, 6 | breqtrri 3950 | 1 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3924 (class class class)co 5767 cc0 7613 c1 7614 caddc 7616 clt 7793 c2 8764 c3 8765 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-2 8772 df-3 8773 |
This theorem is referenced by: 3ne0 8808 3ap0 8809 4pos 8810 8th4div3 8932 halfpm6th 8933 3rp 9440 sqrt9 10813 ef01bndlem 11452 cos2bnd 11456 sin01gt0 11457 cos01gt0 11458 flodddiv4 11620 coseq0negpitopi 12906 tangtx 12908 sincos6thpi 12912 cos02pilt1 12921 ex-gcd 12932 |
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