Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11712 | . . 3 ⊢ 2 ∈ ℝ | |
2 | 1re 10641 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 2pos 11741 | . . 3 ⊢ 0 < 2 | |
4 | 0lt1 11162 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 11182 | . 2 ⊢ 0 < (2 + 1) |
6 | df-3 11702 | . 2 ⊢ 3 = (2 + 1) | |
7 | 5, 6 | breqtrri 5093 | 1 ⊢ 0 < 3 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5066 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 2c2 11693 3c3 11694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-2 11701 df-3 11702 |
This theorem is referenced by: 3ne0 11744 4pos 11745 3rp 12396 fz0to4untppr 13011 s4fv0 14257 sqrlem7 14608 sqrt9 14633 ef01bndlem 15537 cos2bnd 15541 sin01gt0 15543 cos01gt0 15544 rpnnen2lem3 15569 rpnnen2lem4 15570 rpnnen2lem9 15575 flodddiv4 15764 43prm 16455 cnfldfun 20557 tangtx 25091 sincos6thpi 25101 pige3ALT 25105 log2cnv 25522 log2tlbnd 25523 ppiub 25780 bposlem2 25861 bposlem3 25862 bposlem4 25863 bposlem5 25864 lgsdir2lem1 25901 dchrvmasumiflem1 26077 tgcgr4 26317 frgrogt3nreg 28176 friendshipgt3 28177 ex-gcd 28236 cyc3fv3 30781 cyc3conja 30799 hgt750lemd 31919 hgt750lem2 31923 heiborlem5 35108 heiborlem7 35110 jm2.23 39613 stoweidlem13 42318 stoweidlem26 42331 stoweidlem34 42339 stoweidlem42 42347 stoweidlem59 42364 stoweid 42368 wallispilem4 42373 smfmullem4 43089 257prm 43743 127prm 43783 nfermltl2rev 43928 |
Copyright terms: Public domain | W3C validator |