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Theorem snunioc 13433

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 13343	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐴) = {𝐴})
3	2	uneq1d 4107	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = ({𝐴} ∪ (𝐴(,]𝐵)))
4		simp1 1137	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
5		simp2 1138	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
6		xrleid 13102	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
7	6	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
8		simp3 1139	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
9		df-icc 13305	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		df-ioc 13303	. . . 4 ⊢ (,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
11		xrltnle 11212	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
12		xrletr 13109	. . . 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 13101	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤)) → 𝐴 ≤ 𝑤)
17	16	ex 412	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤) → 𝐴 ≤ 𝑤))
18	9, 10, 11, 9, 12, 17	ixxun 13314	. . 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 1381	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐴) ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))
20	3, 19	eqtr3d 2773	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ({𝐴} ∪ (𝐴(,]𝐵)) = (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∪ cun 3887 {csn 4567 class class class wbr 5085 (class class class)co 7367 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 (,]cioc 13299 [,]cicc 13301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-ioc 13303 df-icc 13305

This theorem is referenced by: elntg2 29054 xrge0iifcnv 34077 xrge0iifiso 34079 xrge0iifhom 34081

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