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Mirrors > Home > ILE Home > Th. List > 4pos | Unicode version |
Description: The number 4 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
4pos |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 8969 |
. . 3
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2 | 1re 7934 |
. . 3
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3 | 3pos 8989 |
. . 3
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4 | 0lt1 8061 |
. . 3
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5 | 1, 2, 3, 4 | addgt0ii 8425 |
. 2
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6 | df-4 8956 |
. 2
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7 | 5, 6 | breqtrri 4027 |
1
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Colors of variables: wff set class |
Syntax hints: class class
class wbr 4000 (class class class)co 5868
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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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4628 df-iota 5173 df-fv 5219 df-ov 5871 df-pnf 7971 df-mnf 7972 df-ltxr 7974 df-2 8954 df-3 8955 df-4 8956 |
This theorem is referenced by: 4ne0 8993 4ap0 8994 5pos 8995 8th4div3 9114 div4p1lem1div2 9148 fldiv4p1lem1div2 10278 iexpcyc 10597 faclbnd2 10693 resqrexlemover 10990 resqrexlemcalc1 10994 resqrexlemcalc2 10995 resqrexlemcalc3 10996 resqrexlemnm 10998 resqrexlemga 11003 sqrt2gt1lt2 11029 flodddiv4 11909 dveflem 13820 coseq0negpitopi 13890 sincos4thpi 13894 |
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