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Mirrors > Home > ILE Home > Th. List > 7lt9 | GIF version |
Description: 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
7lt9 | ⊢ 7 < 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7lt8 9023 | . 2 ⊢ 7 < 8 | |
2 | 8lt9 9030 | . 2 ⊢ 8 < 9 | |
3 | 7re 8916 | . . 3 ⊢ 7 ∈ ℝ | |
4 | 8re 8918 | . . 3 ⊢ 8 ∈ ℝ | |
5 | 9re 8920 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 7981 | . 2 ⊢ ((7 < 8 ∧ 8 < 9) → 7 < 9) |
7 | 1, 2, 6 | mp2an 423 | 1 ⊢ 7 < 9 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3965 < clt 7912 7c7 8889 8c8 8890 9c9 8891 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-iota 5135 df-fv 5178 df-ov 5827 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-2 8892 df-3 8893 df-4 8894 df-5 8895 df-6 8896 df-7 8897 df-8 8898 df-9 8899 |
This theorem is referenced by: 6lt9 9032 |
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