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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 8020 | . . 3 ⊢ 1 ∈ ℝ | |
2 | 1 | ltp1i 8926 | . 2 ⊢ 1 < (1 + 1) |
3 | df-2 9043 | . 2 ⊢ 2 = (1 + 1) | |
4 | 2, 3 | breqtrri 4057 | 1 ⊢ 1 < 2 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4030 (class class class)co 5919 1c1 7875 + caddc 7877 < clt 8056 2c2 9035 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-pre-ltadd 7990 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-xp 4666 df-iota 5216 df-fv 5263 df-ov 5922 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-2 9043 |
This theorem is referenced by: 1lt3 9156 1lt4 9159 1lt6 9168 1lt7 9174 1lt8 9181 1lt9 9189 1ne2 9191 1ap2 9192 1le2 9193 halflt1 9202 nn0ge2m1nn 9303 nn0n0n1ge2b 9399 halfnz 9416 1lt10 9589 fztpval 10152 ige2m2fzo 10268 wrdlenge2n0 10952 sqrt2gt1lt2 11196 ege2le3 11817 cos12dec 11914 ene1 11931 eap1 11932 n2dvds1 12056 2prm 12268 3prm 12269 4nprm 12270 isprm5 12283 basendxltplusgndx 12734 grpstrg 12746 grpbaseg 12747 grpplusgg 12748 rngstrg 12755 lmodstrd 12784 topgrpstrd 12816 reeff1o 14949 cosz12 14956 2logb9irrALT 15147 sqrt2cxp2logb9e3 15148 lgseisenlem1 15227 |
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