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Mirrors > Home > ILE Home > Th. List > ioodisj | 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 | simpllr 486 | . . . . . 6 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → B ∈ ℝ*) | |
2 | iooss1 8555 | . . . . . 6 ⊢ ((B ∈ ℝ* ∧ B ≤ 𝐶) → (𝐶(,)𝐷) ⊆ (B(,)𝐷)) | |
3 | 1, 2 | sylancom 397 | . . . . 5 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → (𝐶(,)𝐷) ⊆ (B(,)𝐷)) |
4 | ioossicc 8598 | . . . . 5 ⊢ (B(,)𝐷) ⊆ (B[,]𝐷) | |
5 | 3, 4 | syl6ss 2951 | . . . 4 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → (𝐶(,)𝐷) ⊆ (B[,]𝐷)) |
6 | sslin 3157 | . . . 4 ⊢ ((𝐶(,)𝐷) ⊆ (B[,]𝐷) → ((A(,)B) ∩ (𝐶(,)𝐷)) ⊆ ((A(,)B) ∩ (B[,]𝐷))) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → ((A(,)B) ∩ (𝐶(,)𝐷)) ⊆ ((A(,)B) ∩ (B[,]𝐷))) |
8 | simplll 485 | . . . 4 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → A ∈ ℝ*) | |
9 | simplrr 488 | . . . 4 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → 𝐷 ∈ ℝ*) | |
10 | df-ioo 8531 | . . . . 5 ⊢ (,) = (x ∈ ℝ*, y ∈ ℝ* ↦ {z ∈ ℝ* ∣ (x < z ∧ z < y)}) | |
11 | df-icc 8534 | . . . . 5 ⊢ [,] = (x ∈ ℝ*, y ∈ ℝ* ↦ {z ∈ ℝ* ∣ (x ≤ z ∧ z ≤ y)}) | |
12 | xrlenlt 6881 | . . . . 5 ⊢ ((B ∈ ℝ* ∧ w ∈ ℝ*) → (B ≤ w ↔ ¬ w < B)) | |
13 | 10, 11, 12 | ixxdisj 8542 | . . . 4 ⊢ ((A ∈ ℝ* ∧ B ∈ ℝ* ∧ 𝐷 ∈ ℝ*) → ((A(,)B) ∩ (B[,]𝐷)) = ∅) |
14 | 8, 1, 9, 13 | syl3anc 1134 | . . 3 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → ((A(,)B) ∩ (B[,]𝐷)) = ∅) |
15 | 7, 14 | sseqtrd 2975 | . 2 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → ((A(,)B) ∩ (𝐶(,)𝐷)) ⊆ ∅) |
16 | ss0 3251 | . 2 ⊢ (((A(,)B) ∩ (𝐶(,)𝐷)) ⊆ ∅ → ((A(,)B) ∩ (𝐶(,)𝐷)) = ∅) | |
17 | 15, 16 | syl 14 | 1 ⊢ ((((A ∈ ℝ* ∧ B ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ B ≤ 𝐶) → ((A(,)B) ∩ (𝐶(,)𝐷)) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∩ cin 2910 ⊆ wss 2911 ∅c0 3218 class class class wbr 3755 (class class class)co 5455 ℝ*cxr 6856 < clt 6857 ≤ cle 6858 (,)cioo 8527 [,]cicc 8530 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-po 4024 df-iso 4025 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-ioo 8531 df-icc 8534 |
This theorem is referenced by: (None) |
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