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Theorem snunioc 12514

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 12434	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
2	1	3ad2ant1 1128	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐴) = {𝐴})
3	2	uneq1d 3910	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = ({𝐴} ∪ (𝐴(,]𝐵)))
4		simp1 1131	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
5		simp2 1132	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
6		xrleid 12197	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
7	6	3ad2ant1 1128	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
8		simp3 1133	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
9		df-icc 12396	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		df-ioc 12394	. . . 4 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
11		xrltnle 10318	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
12		xrletr 12203	. . . 4 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝑤 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵) → 𝑤 ≤ 𝐵))
13		simprr 813	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 < 𝑤)
14		simpl1 1228	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ∈ ℝ^)
15		simpl3 1232	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝑤 ∈ ℝ^)
16		xrltle 12196	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
17	14, 15, 16	syl2anc 696	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → (𝐴 < 𝑤 → 𝐴 ≤ 𝑤))
18	13, 17	mpd 15	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ≤ 𝑤)
19	18	ex 449	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤) → 𝐴 ≤ 𝑤))
20	9, 10, 11, 9, 12, 19	ixxun 12405	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
21	4, 4, 5, 7, 8, 20	syl32anc 1485	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
22	3, 21	eqtr3d 2797	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2140 ∪ cun 3714 {csn 4322 class class class wbr 4805 (class class class)co 6815 ℝ^*cxr 10286 < clt 10287 ≤ cle 10288 (,]cioc 12390 [,]cicc 12392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 ax-cnex 10205 ax-resscn 10206 ax-pre-lttri 10223 ax-pre-lttrn 10224

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-nul 4060 df-if 4232 df-pw 4305 df-sn 4323 df-pr 4325 df-op 4329 df-uni 4590 df-br 4806 df-opab 4866 df-mpt 4883 df-id 5175 df-po 5188 df-so 5189 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-ov 6818 df-oprab 6819 df-mpt2 6820 df-er 7914 df-en 8125 df-dom 8126 df-sdom 8127 df-pnf 10289 df-mnf 10290 df-xr 10291 df-ltxr 10292 df-le 10293 df-ioc 12394 df-icc 12396

This theorem is referenced by: xrge0iifcnv 30310 xrge0iifiso 30312 xrge0iifhom 30314

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