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Mirrors > Home > ILE Home > Th. List > 2lt8 | GIF version |
Description: 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
2lt8 | ⊢ 2 < 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt3 9037 | . 2 ⊢ 2 < 3 | |
2 | 3lt8 9061 | . 2 ⊢ 3 < 8 | |
3 | 2re 8937 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 3re 8941 | . . 3 ⊢ 3 ∈ ℝ | |
5 | 8re 8952 | . . 3 ⊢ 8 ∈ ℝ | |
6 | 3, 4, 5 | lttri 8013 | . 2 ⊢ ((2 < 3 ∧ 3 < 8) → 2 < 8) |
7 | 1, 2, 6 | mp2an 424 | 1 ⊢ 2 < 8 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3987 < clt 7943 2c2 8918 3c3 8919 8c8 8924 |
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 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-i2m1 7868 ax-0lt1 7869 ax-0id 7871 ax-rnegex 7872 ax-pre-lttrn 7877 ax-pre-ltadd 7879 |
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 7945 df-mnf 7946 df-ltxr 7948 df-2 8926 df-3 8927 df-4 8928 df-5 8929 df-6 8930 df-7 8931 df-8 8932 |
This theorem is referenced by: 1lt8 9063 |
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