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Mirrors > Home > ILE Home > Th. List > 6pos | GIF version |
Description: The number 6 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
6pos | ⊢ 0 < 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5re 8950 | . . 3 ⊢ 5 ∈ ℝ | |
2 | 1re 7912 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 5pos 8971 | . . 3 ⊢ 0 < 5 | |
4 | 0lt1 8039 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 8403 | . 2 ⊢ 0 < (5 + 1) |
6 | df-6 8934 | . 2 ⊢ 6 = (5 + 1) | |
7 | 5, 6 | breqtrri 4014 | 1 ⊢ 0 < 6 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3987 (class class class)co 5851 0cc0 7767 1c1 7768 + caddc 7770 < clt 7947 5c5 8925 6c6 8926 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-iota 5158 df-fv 5204 df-ov 5854 df-pnf 7949 df-mnf 7950 df-ltxr 7952 df-2 8930 df-3 8931 df-4 8932 df-5 8933 df-6 8934 |
This theorem is referenced by: 7pos 8973 8th4div3 9090 halfpm6th 9091 5recm6rec 9479 efi4p 11673 resin4p 11674 recos4p 11675 ef01bndlem 11712 sin01bnd 11713 cos01bnd 11714 sincos6thpi 13522 pigt3 13524 |
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