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Mirrors > Home > ILE Home > Th. List > 2ap0 | Unicode version |
Description: The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.) |
Ref | Expression |
---|---|
2ap0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 8814 |
. 2
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2 | 2pos 8835 |
. 2
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3 | 1, 2 | gt0ap0ii 8414 |
1
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Colors of variables: wff set class |
Syntax hints: class class
class wbr 3937 ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-2 8803 |
This theorem is referenced by: 2div2e1 8876 4d2e2 8904 halfre 8957 1mhlfehlf 8962 halfpm6th 8964 2muliap0 8968 halfcl 8970 rehalfcl 8971 half0 8972 2halves 8973 halfaddsub 8978 xp1d2m1eqxm1d2 8996 div4p1lem1div2 8997 zneo 9176 nneoor 9177 nneo 9178 zeo 9180 zeo2 9181 halfthird 9348 qbtwnrelemcalc 10064 2tnp1ge0ge0 10105 zesq 10441 sqoddm1div8 10475 faclbnd2 10520 crre 10661 addcj 10695 resqrexlemover 10814 resqrexlemcalc1 10818 resqrexlemcvg 10823 maxabslemab 11010 max0addsup 11023 minabs 11039 bdtri 11043 arisum 11299 arisum2 11300 geo2sum 11315 geo2lim 11317 geoihalfsum 11323 ege2le3 11414 efgt0 11427 tanval2ap 11456 tanval3ap 11457 efi4p 11460 efival 11475 cosadd 11480 sinmul 11487 cosmul 11488 sin01bnd 11500 cos01bnd 11501 sin02gt0 11506 odd2np1 11606 mulsucdiv2z 11618 ltoddhalfle 11626 halfleoddlt 11627 nn0enne 11635 nn0o 11640 flodddiv4 11667 flodddiv4t2lthalf 11670 6lcm4e12 11804 sqrt2irrlem 11875 sqrt2irr 11876 oddennn 11941 evenennn 11942 coscn 12899 sinhalfpilem 12920 cospi 12929 ptolemy 12953 sincosq3sgn 12957 sincosq4sgn 12958 sinq12gt0 12959 cosq23lt0 12962 coseq00topi 12964 tangtx 12967 sincos4thpi 12969 sincos6thpi 12971 sincos3rdpi 12972 pigt3 12973 abssinper 12975 coskpi 12977 logsqrt 13051 apdifflemr 13415 apdiff 13416 |
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