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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 12336 | . . 3 ⊢ 7 ∈ ℝ | |
2 | 1re 11245 | . . 3 ⊢ 1 ∈ ℝ | |
3 | 7pos 12354 | . . 3 ⊢ 0 < 7 | |
4 | 0lt1 11767 | . . 3 ⊢ 0 < 1 | |
5 | 1, 2, 3, 4 | addgt0ii 11787 | . 2 ⊢ 0 < (7 + 1) |
6 | df-8 12312 | . 2 ⊢ 8 = (7 + 1) | |
7 | 5, 6 | breqtrri 5175 | 1 ⊢ 0 < 8 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5148 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 < clt 11279 7c7 12303 8c8 12304 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 |
This theorem is referenced by: 9pos 12356 8th4div3 12463 chtub 27158 bposlem8 27237 bposlem9 27238 lgsdir2lem1 27271 lgsdir2lem4 27274 lgsdir2lem5 27275 2lgsoddprmlem1 27354 2lgsoddprmlem2 27355 2lgsoddprmlem3a 27356 2lgsoddprmlem3b 27357 2lgsoddprmlem3c 27358 2lgsoddprmlem3d 27359 chebbnd1lem2 27416 chebbnd1lem3 27417 pntlemf 27551 hgt750lem 34283 lcmineqlem23 41522 aks4d1p1 41547 imsqrtvalex 43076 fmtnoprmfac2lem1 46906 |
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