ioounsn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ioounsn		Structured version Visualization version GIF version

Theorem ioounsn 13425

Description: The union of an open interval with its upper endpoint is a left-open right-closed interval. (Contributed by Jon Pennant, 8-Jun-2019.)

**Assertion**
Ref	Expression
ioounsn	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))

Proof of Theorem ioounsn Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1144	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^*)
2		iccid 13338	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐵[,]𝐵) = {𝐵})
4	3	uneq2d 4100	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ (𝐵[,]𝐵)) = ((𝐴(,)𝐵) ∪ {𝐵}))
5		simp1 1143	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^*)
6		simp3 1145	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
7	1	xrleidd 13098	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵)
8		df-ioo 13297	. . . 4 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
9		df-icc 13300	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		xrlenlt 11206	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
11		df-ioc 13298	. . . 4 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
12		simpl1 1199	. . . . . 6 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝑤 ∈ ℝ^)
13		simpl2 1200	. . . . . 6 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝐵 ∈ ℝ^)
14		simprl 777	. . . . . 6 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝑤 < 𝐵)
15	12, 13, 14	xrltled 13096	. . . . 5 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝑤 ≤ 𝐵)
16	15	ex 414	. . . 4 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵) → 𝑤 ≤ 𝐵))
17		xrltletr 13103	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤) → 𝐴 < 𝑤))
18	8, 9, 10, 11, 16, 17	ixxun 13309	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → ((𝐴(,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴(,]𝐵))
19	5, 1, 1, 6, 7, 18	syl32anc 1387	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴(,]𝐵))
20	4, 19	eqtr3d 2778	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∪ cun 3882 {csn 4557 class class class wbr 5074 (class class class)co 7359 ℝ^*cxr 11174 < clt 11175 ≤ cle 11176 (,)cioo 13293 (,]cioc 13294 [,]cicc 13296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-pre-lttri 11108 ax-pre-lttrn 11109

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-po 5528 df-so 5529 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-ov 7362 df-oprab 7363 df-mpo 7364 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-ioo 13297 df-ioc 13298 df-icc 13300

This theorem is referenced by: iocunico 43669 iocmbl 43671 limcicciooub 46092 limcresiooub 46097 ioccncflimc 46340 volioc 46427 fourierdlem33 46595 fourierdlem49 46610 fourierdlem93 46654 fouriersw 46686

W3C validator