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Mirrors > Home > ILE Home > Th. List > 1lt2 | GIF version |
Description: 1 is less than 2. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
1lt2 | ⊢ 1 < 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7733 | . . 3 ⊢ 1 ∈ ℝ | |
2 | 1 | ltp1i 8631 | . 2 ⊢ 1 < (1 + 1) |
3 | df-2 8747 | . 2 ⊢ 2 = (1 + 1) | |
4 | 2, 3 | breqtrri 3925 | 1 ⊢ 1 < 2 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3899 (class class class)co 5742 1c1 7589 + caddc 7591 < clt 7768 2c2 8739 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-2 8747 |
This theorem is referenced by: 1lt3 8859 1lt4 8862 1lt6 8871 1lt7 8877 1lt8 8884 1lt9 8892 1ne2 8894 1ap2 8895 1le2 8896 halflt1 8905 nn0ge2m1nn 9005 nn0n0n1ge2b 9098 halfnz 9115 1lt10 9288 fztpval 9831 ige2m2fzo 9943 sqrt2gt1lt2 10789 ege2le3 11304 cos12dec 11401 ene1 11418 eap1 11419 n2dvds1 11536 2prm 11735 3prm 11736 4nprm 11737 grpstrg 11993 grpbaseg 11994 grpplusgg 11995 rngstrg 12001 lmodstrd 12019 topgrpstrd 12037 cosz12 12788 |
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