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Mirrors > Home > MPE Home > Th. List > neg1lt0 | Structured version Visualization version GIF version |
Description: -1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
neg1lt0 | ⊢ -1 < 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg0 10921 | . . 3 ⊢ -0 = 0 | |
2 | 0lt1 11151 | . . 3 ⊢ 0 < 1 | |
3 | 1, 2 | eqbrtri 5051 | . 2 ⊢ -0 < 1 |
4 | 1re 10630 | . . 3 ⊢ 1 ∈ ℝ | |
5 | 0re 10632 | . . 3 ⊢ 0 ∈ ℝ | |
6 | 4, 5 | ltnegcon1i 11180 | . 2 ⊢ (-1 < 0 ↔ -0 < 1) |
7 | 3, 6 | mpbir 234 | 1 ⊢ -1 < 0 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5030 0cc0 10526 1c1 10527 < clt 10664 -cneg 10860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: risefall0lem 15372 binomfallfaclem2 15386 nthruz 15598 psgnodpmr 20279 xrhmph 23552 vitalilem4 24215 vitali 24217 atanre 25471 lgsdir2lem3 25911 ballotlem1c 31875 sgnnbi 31913 sgnpbi 31914 sgnsgn 31916 sgnmulsgn 31917 signswch 31941 fz0n 33075 bcneg1 33081 cnndvlem1 33989 irrdiff 34740 asindmre 35140 stoweidlem7 42649 stirlinglem6 42721 fouriersw 42873 dignn0flhalflem1 45029 |
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