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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 8991 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1re 7955 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 3pos 9011 | . . 3 ⊢ 0 < 3 | |
4 | 0lt1 8082 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 8446 | . 2 ⊢ 0 < (3 + 1) |
6 | df-4 8978 | . 2 ⊢ 4 = (3 + 1) | |
7 | 5, 6 | breqtrri 4030 | 1 ⊢ 0 < 4 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4003 (class class class)co 5874 0cc0 7810 1c1 7811 + caddc 7813 < clt 7990 3c3 8969 4c4 8970 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-iota 5178 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-2 8976 df-3 8977 df-4 8978 |
This theorem is referenced by: 4ne0 9015 4ap0 9016 5pos 9017 8th4div3 9136 div4p1lem1div2 9170 fldiv4p1lem1div2 10302 iexpcyc 10621 faclbnd2 10717 resqrexlemover 11014 resqrexlemcalc1 11018 resqrexlemcalc2 11019 resqrexlemcalc3 11020 resqrexlemnm 11022 resqrexlemga 11027 sqrt2gt1lt2 11053 flodddiv4 11933 dveflem 14080 coseq0negpitopi 14150 sincos4thpi 14154 |
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