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Mirrors > Home > ILE Home > Th. List > 2lt10 | GIF version |
Description: 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
2lt10 | ⊢ 2 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2lt3 9035 | . 2 ⊢ 2 < 3 | |
2 | 3lt10 9466 | . 2 ⊢ 3 < ;10 | |
3 | 2re 8935 | . . 3 ⊢ 2 ∈ ℝ | |
4 | 3re 8939 | . . 3 ⊢ 3 ∈ ℝ | |
5 | 10re 9348 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 8011 | . 2 ⊢ ((2 < 3 ∧ 3 < ;10) → 2 < ;10) |
7 | 1, 2, 6 | mp2an 424 | 1 ⊢ 2 < ;10 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3987 0cc0 7761 1c1 7762 < clt 7941 2c2 8916 3c3 8917 ;cdc 9330 |
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 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
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-int 3830 df-br 3988 df-opab 4049 df-xp 4615 df-iota 5158 df-fv 5204 df-ov 5853 df-pnf 7943 df-mnf 7944 df-ltxr 7946 df-inn 8866 df-2 8924 df-3 8925 df-4 8926 df-5 8927 df-6 8928 df-7 8929 df-8 8930 df-9 8931 df-dec 9331 |
This theorem is referenced by: 1lt10 9468 |
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