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Mirrors > Home > ILE Home > Th. List > 4pos | GIF version |
Description: The number 4 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
4pos | ⊢ 0 < 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 8922 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1re 7889 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 3pos 8942 | . . 3 ⊢ 0 < 3 | |
4 | 0lt1 8016 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 8380 | . 2 ⊢ 0 < (3 + 1) |
6 | df-4 8909 | . 2 ⊢ 4 = (3 + 1) | |
7 | 5, 6 | breqtrri 4003 | 1 ⊢ 0 < 4 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3976 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 < clt 7924 3c3 8900 4c4 8901 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-iota 5147 df-fv 5190 df-ov 5839 df-pnf 7926 df-mnf 7927 df-ltxr 7929 df-2 8907 df-3 8908 df-4 8909 |
This theorem is referenced by: 4ne0 8946 4ap0 8947 5pos 8948 8th4div3 9067 div4p1lem1div2 9101 fldiv4p1lem1div2 10230 iexpcyc 10549 faclbnd2 10644 resqrexlemover 10938 resqrexlemcalc1 10942 resqrexlemcalc2 10943 resqrexlemcalc3 10944 resqrexlemnm 10946 resqrexlemga 10951 sqrt2gt1lt2 10977 flodddiv4 11856 dveflem 13234 coseq0negpitopi 13304 sincos4thpi 13308 |
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