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Theorem snunioc 13068

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 12980	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
2	1	3ad2ant1 1135	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐴) = {𝐴})
3	2	uneq1d 4076	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = ({𝐴} ∪ (𝐴(,]𝐵)))
4		simp1 1138	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
5		simp2 1139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
6		xrleid 12741	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
7	6	3ad2ant1 1135	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
8		simp3 1140	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
9		df-icc 12942	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		df-ioc 12940	. . . 4 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
11		xrltnle 10900	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
12		xrletr 12748	. . . 4 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝑤 ≤ 𝐵))
13		simpl1 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ∈ ℝ^)
14		simpl3 1195	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝑤 ∈ ℝ^)
15		simprr 773	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 < 𝑤)
16	13, 14, 15	xrltled 12740	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ≤ 𝑤)
17	16	ex 416	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤) → 𝐴 ≤ 𝑤))
18	9, 10, 11, 9, 12, 17	ixxun 12951	. . 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 1380	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
20	3, 19	eqtr3d 2779	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∪ cun 3864 {csn 4541 class class class wbr 5053 (class class class)co 7213 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 (,]cioc 12936 [,]cicc 12938

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 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-ioc 12940 df-icc 12942

This theorem is referenced by: elntg2 27076 xrge0iifcnv 31597 xrge0iifiso 31599 xrge0iifhom 31601

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