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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 8952 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1re 7919 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 3pos 8972 | . . 3 ⊢ 0 < 3 | |
4 | 0lt1 8046 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 8410 | . 2 ⊢ 0 < (3 + 1) |
6 | df-4 8939 | . 2 ⊢ 4 = (3 + 1) | |
7 | 5, 6 | breqtrri 4016 | 1 ⊢ 0 < 4 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3989 (class class class)co 5853 0cc0 7774 1c1 7775 + caddc 7777 < clt 7954 3c3 8930 4c4 8931 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-iota 5160 df-fv 5206 df-ov 5856 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-2 8937 df-3 8938 df-4 8939 |
This theorem is referenced by: 4ne0 8976 4ap0 8977 5pos 8978 8th4div3 9097 div4p1lem1div2 9131 fldiv4p1lem1div2 10261 iexpcyc 10580 faclbnd2 10676 resqrexlemover 10974 resqrexlemcalc1 10978 resqrexlemcalc2 10979 resqrexlemcalc3 10980 resqrexlemnm 10982 resqrexlemga 10987 sqrt2gt1lt2 11013 flodddiv4 11893 dveflem 13481 coseq0negpitopi 13551 sincos4thpi 13555 |
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