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Mirrors > Home > MPE Home > Th. List > 8pos | Structured version Visualization version GIF version |
Description: The number 8 is positive. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
8pos | ⊢ 0 < 8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7re 11455 | . . 3 ⊢ 7 ∈ ℝ | |
2 | 1re 10363 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 7pos 11476 | . . 3 ⊢ 0 < 7 | |
4 | 0lt1 10881 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 10901 | . 2 ⊢ 0 < (7 + 1) |
6 | df-8 11427 | . 2 ⊢ 8 = (7 + 1) | |
7 | 5, 6 | breqtrri 4902 | 1 ⊢ 0 < 8 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 4875 (class class class)co 6910 0cc0 10259 1c1 10260 + caddc 10262 < clt 10398 7c7 11418 8c8 11419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 |
This theorem is referenced by: 9pos 11478 8th4div3 11585 chtub 25357 bposlem8 25436 bposlem9 25437 lgsdir2lem1 25470 lgsdir2lem4 25473 lgsdir2lem5 25474 2lgsoddprmlem1 25553 2lgsoddprmlem2 25554 2lgsoddprmlem3a 25555 2lgsoddprmlem3b 25556 2lgsoddprmlem3c 25557 2lgsoddprmlem3d 25558 chebbnd1lem2 25579 chebbnd1lem3 25580 pntlemf 25714 hgt750lem 31274 fmtnoprmfac2lem1 42322 |
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