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Mirrors > Home > ILE Home > Th. List > xlt0neg2 | GIF version |
Description: Extended real version of lt0neg2 8345. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xlt0neg2 | ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7925 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xltneg 9741 | . . 3 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (0 < 𝐴 ↔ -𝑒𝐴 < -𝑒0)) | |
3 | 1, 2 | mpan 421 | . 2 ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < -𝑒0)) |
4 | xneg0 9736 | . . 3 ⊢ -𝑒0 = 0 | |
5 | 4 | breq2i 3974 | . 2 ⊢ (-𝑒𝐴 < -𝑒0 ↔ -𝑒𝐴 < 0) |
6 | 3, 5 | bitrdi 195 | 1 ⊢ (𝐴 ∈ ℝ* → (0 < 𝐴 ↔ -𝑒𝐴 < 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 2128 class class class wbr 3966 0cc0 7733 ℝ*cxr 7912 < clt 7913 -𝑒cxne 9677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-sub 8049 df-neg 8050 df-xneg 9680 |
This theorem is referenced by: (None) |
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