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Mirrors > Home > ILE Home > Th. List > 3pos | GIF version |
Description: The number 3 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
3pos | ⊢ 0 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8797 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1re 7772 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 2pos 8818 | . . 3 ⊢ 0 < 2 | |
4 | 0lt1 7896 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 8260 | . 2 ⊢ 0 < (2 + 1) |
6 | df-3 8787 | . 2 ⊢ 3 = (2 + 1) | |
7 | 5, 6 | breqtrri 3955 | 1 ⊢ 0 < 3 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3929 (class class class)co 5774 0cc0 7627 1c1 7628 + caddc 7630 < clt 7807 2c2 8778 3c3 8779 |
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 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-pre-lttrn 7741 ax-pre-ltadd 7743 |
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 7809 df-mnf 7810 df-ltxr 7812 df-2 8786 df-3 8787 |
This theorem is referenced by: 3ne0 8822 3ap0 8823 4pos 8824 8th4div3 8946 halfpm6th 8947 3rp 9454 sqrt9 10827 ef01bndlem 11470 cos2bnd 11474 sin01gt0 11475 cos01gt0 11476 flodddiv4 11638 coseq0negpitopi 12927 tangtx 12929 sincos6thpi 12933 cos02pilt1 12942 ex-gcd 12953 |
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