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Mirrors > Home > MPE Home > Th. List > lt0neg2 | Structured version Visualization version GIF version |
Description: Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
lt0neg2 | ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 11261 | . . 3 ⊢ 0 ∈ ℝ | |
2 | ltneg 11761 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ -𝐴 < -0)) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < -0)) |
4 | neg0 11553 | . . 3 ⊢ -0 = 0 | |
5 | 4 | breq2i 5156 | . 2 ⊢ (-𝐴 < -0 ↔ -𝐴 < 0) |
6 | 3, 5 | bitrdi 287 | 1 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 class class class wbr 5148 ℝcr 11152 0cc0 11153 < clt 11293 -cneg 11491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: lt0neg2d 11831 elnnz 12621 sincos2sgn 16227 tanord1 26594 tanregt0 26596 relogrn 26618 logi 26644 logneg 26645 asin1 26952 reasinsin 26954 atanbnd 26984 atan1 26986 sgnneg 34522 bj-pinftynminfty 37210 tan2h 37599 asin1half 42366 negpilt0 45231 stoweidlem34 45990 stirlinglem10 46039 fourierdlem103 46165 |
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