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Mirrors > Home > ILE Home > Th. List > 2lt3 | GIF version |
Description: 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
Ref | Expression |
---|---|
2lt3 | ⊢ 2 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8814 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1 | ltp1i 8687 | . 2 ⊢ 2 < (2 + 1) |
3 | df-3 8804 | . 2 ⊢ 3 = (2 + 1) | |
4 | 2, 3 | breqtrri 3963 | 1 ⊢ 2 < 3 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3937 (class class class)co 5782 1c1 7645 + caddc 7647 < clt 7824 2c2 8795 3c3 8796 |
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-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 |
This theorem is referenced by: 1lt3 8915 2lt4 8917 2lt6 8926 2lt7 8932 2lt8 8939 2lt9 8947 3halfnz 9172 2lt10 9343 uzuzle23 9393 uz3m2nn 9395 fztpval 9894 expnass 10429 cos01gt0 11505 3lcm2e6 11874 plusgndxnmulrndx 12111 rngstrg 12113 coseq00topi 12964 coseq0negpitopi 12965 cos02pilt1 12980 2logb9irr 13096 2logb3irr 13098 2logb9irrap 13102 ex-fl 13108 |
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