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Mirrors > Home > ILE Home > Th. List > iooidg | GIF version |
Description: An open interval with identical lower and upper bounds is empty. (Contributed by Jim Kingdon, 29-Mar-2020.) |
Ref | Expression |
---|---|
iooidg | ⊢ (𝐴 ∈ ℝ* → (𝐴(,)𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval 9235 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴(,)𝐴) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)}) | |
2 | 1 | anidms 389 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴(,)𝐴) = {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)}) |
3 | xrltnsym2 9173 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑥 ∈ ℝ*) → ¬ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)) | |
4 | 3 | ralrimiva 2442 | . . 3 ⊢ (𝐴 ∈ ℝ* → ∀𝑥 ∈ ℝ* ¬ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)) |
5 | rabeq0 3298 | . . 3 ⊢ ({𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)} = ∅ ↔ ∀𝑥 ∈ ℝ* ¬ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)) | |
6 | 4, 5 | sylibr 132 | . 2 ⊢ (𝐴 ∈ ℝ* → {𝑥 ∈ ℝ* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐴)} = ∅) |
7 | 2, 6 | eqtrd 2117 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴(,)𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1287 ∈ wcel 1436 ∀wral 2355 {crab 2359 ∅c0 3272 class class class wbr 3814 (class class class)co 5594 ℝ*cxr 7442 < clt 7443 (,)cioo 9215 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-pre-ltirr 7378 ax-pre-lttrn 7380 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-ioo 9219 |
This theorem is referenced by: (None) |
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