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Mirrors > Home > ILE Home > Th. List > 3lt9 | GIF version |
Description: 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.) |
Ref | Expression |
---|---|
3lt9 | ⊢ 3 < 9 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3lt4 8892 | . 2 ⊢ 3 < 4 | |
2 | 4lt9 8921 | . 2 ⊢ 4 < 9 | |
3 | 3re 8794 | . . 3 ⊢ 3 ∈ ℝ | |
4 | 4re 8797 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 9re 8807 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 7868 | . 2 ⊢ ((3 < 4 ∧ 4 < 9) → 3 < 9) |
7 | 1, 2, 6 | mp2an 422 | 1 ⊢ 3 < 9 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3929 < clt 7800 3c3 8772 4c4 8773 9c9 8778 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 |
This theorem is referenced by: 2lt9 8923 |
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