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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 11901 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1re 10830 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 2pos 11930 | . . 3 ⊢ 0 < 2 | |
4 | 0lt1 11351 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 11371 | . 2 ⊢ 0 < (2 + 1) |
6 | df-3 11891 | . 2 ⊢ 3 = (2 + 1) | |
7 | 5, 6 | breqtrri 5077 | 1 ⊢ 0 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5050 (class class class)co 7210 0cc0 10726 1c1 10727 + caddc 10729 < clt 10864 2c2 11882 3c3 11883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-po 5465 df-so 5466 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-2 11890 df-3 11891 |
This theorem is referenced by: 3ne0 11933 4pos 11934 3rp 12589 fz0to4untppr 13212 s4fv0 14457 sqrlem7 14809 sqrt9 14834 ef01bndlem 15742 cos2bnd 15746 sin01gt0 15748 cos01gt0 15749 rpnnen2lem3 15774 rpnnen2lem4 15775 rpnnen2lem9 15780 flodddiv4 15971 43prm 16672 cnfldfun 20372 tangtx 25392 sincos6thpi 25402 pige3ALT 25406 log2cnv 25824 log2tlbnd 25825 ppiub 26082 bposlem2 26163 bposlem3 26164 bposlem4 26165 bposlem5 26166 lgsdir2lem1 26203 dchrvmasumiflem1 26379 tgcgr4 26619 frgrogt3nreg 28477 friendshipgt3 28478 ex-gcd 28537 cyc3fv3 31122 cyc3conja 31140 hgt750lemd 32337 hgt750lem2 32341 heiborlem5 35708 heiborlem7 35710 3lexlogpow5ineq2 39795 3lexlogpow5ineq4 39796 3lexlogpow5ineq3 39797 3lexlogpow2ineq1 39798 3lexlogpow2ineq2 39799 3lexlogpow5ineq5 39800 aks4d1p1p6 39812 aks4d1p1p5 39814 aks4d1p1 39815 aks4d1p2 39816 aks4d1p3 39817 acos1half 39890 jm2.23 40519 stoweidlem13 43227 stoweidlem26 43240 stoweidlem34 43248 stoweidlem42 43256 stoweidlem59 43273 stoweid 43277 wallispilem4 43282 smfmullem4 43998 257prm 44684 127prm 44722 nfermltl2rev 44866 sepfsepc 45892 |
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