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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 9086 | . 2 ⊢ 3 < 4 | |
2 | 4lt9 9115 | . 2 ⊢ 4 < 9 | |
3 | 3re 8988 | . . 3 ⊢ 3 ∈ ℝ | |
4 | 4re 8991 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 9re 9001 | . . 3 ⊢ 9 ∈ ℝ | |
6 | 3, 4, 5 | lttri 8057 | . 2 ⊢ ((3 < 4 ∧ 4 < 9) → 3 < 9) |
7 | 1, 2, 6 | mp2an 426 | 1 ⊢ 3 < 9 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4002 < clt 7987 3c3 8966 4c4 8967 9c9 8972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-addcom 7907 ax-addass 7909 ax-i2m1 7912 ax-0lt1 7913 ax-0id 7915 ax-rnegex 7916 ax-pre-lttrn 7921 ax-pre-ltadd 7923 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-iota 5176 df-fv 5222 df-ov 5874 df-pnf 7989 df-mnf 7990 df-ltxr 7992 df-2 8973 df-3 8974 df-4 8975 df-5 8976 df-6 8977 df-7 8978 df-8 8979 df-9 8980 |
This theorem is referenced by: 2lt9 9117 tsetndxnmulrndx 12639 |
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