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Mirrors > Home > ILE Home > Th. List > xlt0neg1 | GIF version |
Description: Extended real version of lt0neg1 7848. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xlt0neg1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7436 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xltneg 9192 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 < 0 ↔ -𝑒0 < -𝑒𝐴)) | |
3 | 1, 2 | mpan2 416 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ -𝑒0 < -𝑒𝐴)) |
4 | xneg0 9187 | . . 3 ⊢ -𝑒0 = 0 | |
5 | 4 | breq1i 3818 | . 2 ⊢ (-𝑒0 < -𝑒𝐴 ↔ 0 < -𝑒𝐴) |
6 | 3, 5 | syl6bb 194 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 < 0 ↔ 0 < -𝑒𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∈ wcel 1434 class class class wbr 3811 0cc0 7252 ℝ*cxr 7423 < clt 7424 -𝑒cxne 9134 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-addcom 7347 ax-addass 7349 ax-distr 7351 ax-i2m1 7352 ax-0id 7355 ax-rnegex 7356 ax-cnre 7358 ax-pre-ltadd 7363 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-sub 7557 df-neg 7558 df-xneg 9137 |
This theorem is referenced by: (None) |
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