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Mirrors > Home > ILE Home > Th. List > 4lt8 | GIF version |
Description: 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
4lt8 | ⊢ 4 < 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4lt5 9024 | . 2 ⊢ 4 < 5 | |
2 | 5lt8 9041 | . 2 ⊢ 5 < 8 | |
3 | 4re 8926 | . . 3 ⊢ 4 ∈ ℝ | |
4 | 5re 8928 | . . 3 ⊢ 5 ∈ ℝ | |
5 | 8re 8934 | . . 3 ⊢ 8 ∈ ℝ | |
6 | 3, 4, 5 | lttri 7995 | . 2 ⊢ ((4 < 5 ∧ 5 < 8) → 4 < 8) |
7 | 1, 2, 6 | mp2an 423 | 1 ⊢ 4 < 8 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3977 < clt 7925 4c4 8902 5c5 8903 8c8 8906 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-xp 4605 df-iota 5148 df-fv 5191 df-ov 5840 df-pnf 7927 df-mnf 7928 df-ltxr 7930 df-2 8908 df-3 8909 df-4 8910 df-5 8911 df-6 8912 df-7 8913 df-8 8914 |
This theorem is referenced by: 3lt8 9043 |
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