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Theorem snunioc 13431

Description: The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)

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

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

Step	Hyp	Ref	Expression
1		iccid 13341	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
2	1	3ad2ant1 1139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐴) = {𝐴})
3	2	uneq1d 4104	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = ({𝐴} ∪ (𝐴(,]𝐵)))
4		simp1 1142	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
5		simp2 1143	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
6		xrleid 13100	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
7	6	3ad2ant1 1139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
8		simp3 1144	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
9		df-icc 13303	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		df-ioc 13301	. . . 4 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
11		xrltnle 11210	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
12		xrletr 13107	. . . 4 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝑤 ≤ 𝐵))
13		simpl1 1198	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ∈ ℝ^)
14		simpl3 1200	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝑤 ∈ ℝ^)
15		simprr 778	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 < 𝑤)
16	13, 14, 15	xrltled 13099	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ≤ 𝑤)
17	16	ex 413	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤) → 𝐴 ≤ 𝑤))
18	9, 10, 11, 9, 12, 17	ixxun 13312	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
19	4, 4, 5, 7, 8, 18	syl32anc 1386	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
20	3, 19	eqtr3d 2777	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∪ cun 3888 {csn 4562 class class class wbr 5079 (class class class)co 7363 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 (,]cioc 13297 [,]cicc 13299

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-pre-lttri 11110 ax-pre-lttrn 11111

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-ioc 13301 df-icc 13303

This theorem is referenced by: elntg2 29079 xrge0iifcnv 34124 xrge0iifiso 34126 xrge0iifhom 34128

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