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Mirrors > Home > ILE Home > Th. List > halflt1 | GIF version |
Description: One-half is less than one. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
halflt1 | ⊢ (1 / 2) < 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1div1e1 8621 | . . 3 ⊢ (1 / 1) = 1 | |
2 | 1lt2 9047 | . . 3 ⊢ 1 < 2 | |
3 | 1, 2 | eqbrtri 4010 | . 2 ⊢ (1 / 1) < 2 |
4 | 1re 7919 | . . 3 ⊢ 1 ∈ ℝ | |
5 | 2re 8948 | . . 3 ⊢ 2 ∈ ℝ | |
6 | 0lt1 8046 | . . 3 ⊢ 0 < 1 | |
7 | 2pos 8969 | . . 3 ⊢ 0 < 2 | |
8 | 4, 4, 5, 6, 7 | ltdiv23ii 8843 | . 2 ⊢ ((1 / 1) < 2 ↔ (1 / 2) < 1) |
9 | 3, 8 | mpbi 144 | 1 ⊢ (1 / 2) < 1 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3989 (class class class)co 5853 1c1 7775 < clt 7954 / cdiv 8589 2c2 8929 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
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-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-2 8937 |
This theorem is referenced by: 2tnp1ge0ge0 10257 resqrexlemlo 10977 geo2sum 11477 geo2lim 11479 geoihalfsum 11485 efcllemp 11621 cos12dec 11730 ltoddhalfle 11852 halfleoddlt 11853 cvgcmp2nlemabs 14064 trilpolemisumle 14070 |
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