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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 10961 | . . 3 ⊢ 0 ∈ ℝ | |
2 | ltneg 11458 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐴 ↔ -𝐴 < -0)) | |
3 | 1, 2 | mpan 686 | . 2 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < -0)) |
4 | neg0 11250 | . . 3 ⊢ -0 = 0 | |
5 | 4 | breq2i 5086 | . 2 ⊢ (-𝐴 < -0 ↔ -𝐴 < 0) |
6 | 3, 5 | bitrdi 286 | 1 ⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2109 class class class wbr 5078 ℝcr 10854 0cc0 10855 < clt 10993 -cneg 11189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 |
This theorem is referenced by: lt0neg2d 11528 elnnz 12312 sincos2sgn 15884 tanord1 25674 tanregt0 25676 relogrn 25698 logneg 25724 asin1 26025 reasinsin 26027 atanbnd 26057 atan1 26059 sgnneg 32486 logi 33679 bj-pinftynminfty 35377 tan2h 35748 negpilt0 42772 stoweidlem34 43529 stirlinglem10 43578 fourierdlem103 43704 |
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