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Mirrors > Home > MPE Home > Th. List > 3pos | Structured version Visualization version GIF version |
Description: The number 3 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
3pos | ⊢ 0 < 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 12329 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1re 11252 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 2pos 12358 | . . 3 ⊢ 0 < 2 | |
4 | 0lt1 11774 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 11794 | . 2 ⊢ 0 < (2 + 1) |
6 | df-3 12319 | . 2 ⊢ 3 = (2 + 1) | |
7 | 5, 6 | breqtrri 5170 | 1 ⊢ 0 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5143 (class class class)co 7413 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 2c2 12310 3c3 12311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-2 12318 df-3 12319 |
This theorem is referenced by: 3ne0 12361 4pos 12362 3rp 13025 fz0to4untppr 13649 s4fv0 14896 01sqrexlem7 15245 sqrt9 15270 ef01bndlem 16178 cos2bnd 16182 sin01gt0 16184 cos01gt0 16185 rpnnen2lem3 16210 rpnnen2lem4 16211 rpnnen2lem9 16216 flodddiv4 16407 43prm 17116 slotsdifunifndx 17407 cnfldfunALTOLDOLD 21365 tangtx 26527 sincos6thpi 26537 pige3ALT 26541 log2cnv 26966 log2tlbnd 26967 ppiub 27227 bposlem2 27308 bposlem3 27309 bposlem4 27310 bposlem5 27311 lgsdir2lem1 27348 dchrvmasumiflem1 27524 tgcgr4 28452 frgrogt3nreg 30324 friendshipgt3 30325 ex-gcd 30384 cyc3fv3 33018 cyc3conja 33036 evl1deg3 33453 2sqr3minply 33617 hgt750lemd 34504 hgt750lem2 34508 heiborlem5 37526 heiborlem7 37528 3lexlogpow5ineq2 41764 3lexlogpow5ineq4 41765 3lexlogpow5ineq3 41766 3lexlogpow2ineq1 41767 3lexlogpow2ineq2 41768 3lexlogpow5ineq5 41769 aks4d1lem1 41771 aks4d1p1p6 41782 aks4d1p1p5 41784 aks4d1p1 41785 aks4d1p2 41786 aks4d1p3 41787 aks4d1p5 41789 aks4d1p6 41790 aks4d1p7d1 41791 aks4d1p7 41792 aks4d1p8 41796 aks4d1p9 41797 aks6d1c7lem1 41889 aks6d1c7lem2 41890 aks6d1c7 41893 aks5lem6 41901 acos1half 42363 jm2.23 42688 stoweidlem13 45667 stoweidlem26 45680 stoweidlem34 45688 stoweidlem42 45696 stoweidlem59 45713 stoweid 45717 wallispilem4 45722 smfmullem4 46448 257prm 47166 127prm 47204 nfermltl2rev 47348 sepfsepc 48294 |
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