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Mirrors > Home > ILE Home > Th. List > 2pos | GIF version |
Description: The number 2 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
2pos | ⊢ 0 < 2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7118 | . . 3 ⊢ 1 ∈ ℝ | |
2 | 0lt1 7236 | . . 3 ⊢ 0 < 1 | |
3 | 1, 1, 2, 2 | addgt0ii 7592 | . 2 ⊢ 0 < (1 + 1) |
4 | df-2 8098 | . 2 ⊢ 2 = (1 + 1) | |
5 | 3, 4 | breqtrri 3810 | 1 ⊢ 0 < 2 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3785 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 < clt 7153 2c2 8089 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-iota 4887 df-fv 4930 df-ov 5535 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-2 8098 |
This theorem is referenced by: 2ne0 8131 2ap0 8132 3pos 8133 halfgt0 8246 halflt1 8248 halfpos2 8261 halfnneg2 8263 nominpos 8268 avglt1 8269 avglt2 8270 nn0n0n1ge2b 8427 3halfnz 8444 2rp 8739 2tnp1ge0ge0 9303 mulp1mod1 9367 amgm2 10004 oexpneg 10276 oddge22np1 10281 evennn02n 10282 nn0ehalf 10303 nno 10306 nn0oddm1d2 10309 nnoddm1d2 10310 flodddiv4t2lthalf 10337 sqrt2re 10542 sqrt2irrap 10558 ex-fl 10563 |
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