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Mirrors > Home > ILE Home > Th. List > 7pos | GIF version |
Description: The number 7 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
7pos | ⊢ 0 < 7 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6re 8806 | . . 3 ⊢ 6 ∈ ℝ | |
2 | 1re 7770 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 6pos 8826 | . . 3 ⊢ 0 < 6 | |
4 | 0lt1 7894 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 8258 | . 2 ⊢ 0 < (6 + 1) |
6 | df-7 8789 | . 2 ⊢ 7 = (6 + 1) | |
7 | 5, 6 | breqtrri 3955 | 1 ⊢ 0 < 7 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3929 (class class class)co 5774 0cc0 7625 1c1 7626 + caddc 7628 < clt 7805 6c6 8780 7c7 8781 |
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 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-addcom 7725 ax-addass 7727 ax-i2m1 7730 ax-0lt1 7731 ax-0id 7733 ax-rnegex 7734 ax-pre-lttrn 7739 ax-pre-ltadd 7741 |
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 7807 df-mnf 7808 df-ltxr 7810 df-2 8784 df-3 8785 df-4 8786 df-5 8787 df-6 8788 df-7 8789 |
This theorem is referenced by: 8pos 8828 |
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