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Mirrors > Home > MPE Home > Th. List > ioodisj | Structured version Visualization version GIF version |
Description: If the upper bound of one open interval is less than or equal to the lower bound of the other, the intervals are disjoint. (Contributed by Jeff Hankins, 13-Jul-2009.) |
Ref | Expression |
---|---|
ioodisj | ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooss1 13419 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐶) → (𝐶(,)𝐷) ⊆ (𝐵(,)𝐷)) | |
2 | 1 | ad4ant24 754 | . . . . 5 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → (𝐶(,)𝐷) ⊆ (𝐵(,)𝐷)) |
3 | ioossicc 13470 | . . . . 5 ⊢ (𝐵(,)𝐷) ⊆ (𝐵[,]𝐷) | |
4 | 2, 3 | sstrdi 4008 | . . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → (𝐶(,)𝐷) ⊆ (𝐵[,]𝐷)) |
5 | sslin 4251 | . . . 4 ⊢ ((𝐶(,)𝐷) ⊆ (𝐵[,]𝐷) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ⊆ ((𝐴(,)𝐵) ∩ (𝐵[,]𝐷))) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ⊆ ((𝐴(,)𝐵) ∩ (𝐵[,]𝐷))) |
7 | simplll 775 | . . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ*) | |
8 | simpllr 776 | . . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → 𝐵 ∈ ℝ*) | |
9 | simplrr 778 | . . . 4 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → 𝐷 ∈ ℝ*) | |
10 | df-ioo 13388 | . . . . 5 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) | |
11 | df-icc 13391 | . . . . 5 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
12 | xrlenlt 11324 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵)) | |
13 | 10, 11, 12 | ixxdisj 13399 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → ((𝐴(,)𝐵) ∩ (𝐵[,]𝐷)) = ∅) |
14 | 7, 8, 9, 13 | syl3anc 1370 | . . 3 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐵[,]𝐷)) = ∅) |
15 | 6, 14 | sseqtrd 4036 | . 2 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ⊆ ∅) |
16 | ss0 4408 | . 2 ⊢ (((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) ⊆ ∅ → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅) | |
17 | 15, 16 | syl 17 | 1 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 (class class class)co 7431 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 [,]cicc 13387 |
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-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
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-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-iun 4998 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-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 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-ioo 13388 df-icc 13391 |
This theorem is referenced by: reconnlem1 24862 dyaddisjlem 25644 itgsplitioo 25888 |
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