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 12963

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 1138	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^*)
2		iccid 12878	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐵[,]𝐵) = {𝐵})
4	3	uneq2d 4063	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ (𝐵[,]𝐵)) = ((𝐴(,)𝐵) ∪ {𝐵}))
5		simp1 1137	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^*)
6		simp3 1139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
7	1	xrleidd 12640	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵)
8		df-ioo 12837	. . . 4 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
9		df-icc 12840	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		xrlenlt 10796	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
11		df-ioc 12838	. . . 4 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
12		simpl1 1192	. . . . . 6 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝑤 ∈ ℝ^)
13		simpl2 1193	. . . . . 6 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝐵 ∈ ℝ^)
14		simprl 771	. . . . . 6 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝑤 < 𝐵)
15	12, 13, 14	xrltled 12638	. . . . 5 ⊢ (((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵)) → 𝑤 ≤ 𝐵)
16	15	ex 416	. . . 4 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵) → 𝑤 ≤ 𝐵))
17		xrltletr 12645	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤) → 𝐴 < 𝑤))
18	8, 9, 10, 11, 16, 17	ixxun 12849	. . 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 1379	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴(,]𝐵))
20	4, 19	eqtr3d 2776	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∪ cun 3851 {csn 4526 class class class wbr 5040 (class class class)co 7182 ℝ^*cxr 10764 < clt 10765 ≤ cle 10766 (,)cioo 12833 (,]cioc 12834 [,]cicc 12836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-pre-lttri 10701 ax-pre-lttrn 10702

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-ioo 12837 df-ioc 12838 df-icc 12840

This theorem is referenced by: iocunico 40654 iocmbl 40656 limcicciooub 42760 limcresiooub 42765 ioccncflimc 43008 volioc 43095 fourierdlem33 43263 fourierdlem49 43278 fourierdlem93 43322 fouriersw 43354

W3C validator