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Theorem snunico 13416

Description: The closure of the open end of a right-open real interval. (Contributed by Mario Carneiro, 16-Jun-2014.)

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

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

Step	Hyp	Ref	Expression
1		simp2 1137	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
2		iccid 13327	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵[,]𝐵) = {𝐵})
4	3	uneq2d 4127	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,]𝐵)) = ((𝐴[,)𝐵) ∪ {𝐵}))
5		simp1 1136	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
6		simp3 1138	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
7	1	xrleidd 13088	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
8		df-ico 13288	. . . 4 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
9		df-icc 13289	. . . 4 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
10		xrlenlt 11215	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
11		xrltle 13085	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
12	11	3adant3 1132	. . . . 5 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 < 𝐵 → 𝑤 ≤ 𝐵))
13	12	adantrd 491	. . . 4 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵) → 𝑤 ≤ 𝐵))
14		xrletr 13094	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝑤) → 𝐴 ≤ 𝑤))
15	8, 9, 10, 9, 13, 14	ixxun 13298	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)) → ((𝐴[,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴[,]𝐵))
16	5, 1, 1, 6, 7, 15	syl32anc 1380	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ (𝐵[,]𝐵)) = (𝐴[,]𝐵))
17	4, 16	eqtr3d 2766	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,)𝐵) ∪ {𝐵}) = (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∪ cun 3909 {csn 4585 class class class wbr 5102 (class class class)co 7369 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 [,)cico 13284 [,]cicc 13285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-pre-lttri 11118 ax-pre-lttrn 11119

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 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 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-ico 13288 df-icc 13289

This theorem is referenced by: prunioo 13418 iccpnfcnv 24818 iccpnfhmeo 24819 elntg2 28888 xrge0iifcnv 33896

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