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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 8916 | . 2 ⊢ 3 < 4 | |
2 | 4lt9 8945 | . 2 ⊢ 4 < 9 | |
3 | 3re 8818 | . . 3 ⊢ 3 ∈ ℝ | |
4 | 4re 8821 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 9re 8831 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 7892 | . 2 ⊢ ((3 < 4 ∧ 4 < 9) → 3 < 9) |
7 | 1, 2, 6 | mp2an 423 | 1 ⊢ 3 < 9 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3937 < clt 7824 3c3 8796 4c4 8797 9c9 8802 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 |
This theorem is referenced by: 2lt9 8947 |
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