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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 8590 |
. 2
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2 | 2pos 8611 |
. 2
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3 | 1, 2 | gt0ap0ii 8201 |
1
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Colors of variables: wff set class |
Syntax hints: class class
class wbr 3867 ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-ltxr 7624 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-2 8579 |
This theorem is referenced by: 2div2e1 8646 4d2e2 8674 halfre 8727 1mhlfehlf 8732 halfpm6th 8734 2muliap0 8738 halfcl 8740 rehalfcl 8741 half0 8742 2halves 8743 halfaddsub 8748 xp1d2m1eqxm1d2 8766 div4p1lem1div2 8767 zneo 8946 nneoor 8947 nneo 8948 zeo 8950 zeo2 8951 qbtwnrelemcalc 9816 2tnp1ge0ge0 9857 zesq 10187 sqoddm1div8 10221 faclbnd2 10265 crre 10406 addcj 10440 resqrexlemover 10558 resqrexlemcalc1 10562 resqrexlemcvg 10567 maxabslemab 10754 max0addsup 10767 minabs 10782 bdtri 10786 arisum 11041 arisum2 11042 geo2sum 11057 geo2lim 11059 geoihalfsum 11065 ege2le3 11110 efgt0 11123 tanval2ap 11153 tanval3ap 11154 efi4p 11157 efival 11172 cosadd 11177 sinmul 11184 cosmul 11185 sin01bnd 11197 cos01bnd 11198 sin02gt0 11203 odd2np1 11300 mulsucdiv2z 11312 ltoddhalfle 11320 halfleoddlt 11321 nn0enne 11329 nn0o 11334 flodddiv4 11361 flodddiv4t2lthalf 11364 6lcm4e12 11496 sqrt2irrlem 11567 sqrt2irr 11568 oddennn 11632 evenennn 11633 |
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