ixxssixx - Metamath Proof Explorer

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Theorem ixxssixx 13093

Description: An interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixx.2	⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})
ixx.3	⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤))
ixx.4	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))

**Assertion**
Ref	Expression
ixxssixx	⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐴 𝑤,𝑂 𝑤,𝐵,𝑥,𝑦,𝑧 𝑤,𝑃 𝑥,𝑅,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝑥,𝑇,𝑦,𝑧 𝑥,𝑈,𝑦,𝑧 Allowed substitution hints: 𝑃(𝑥,𝑦,𝑧) 𝑅(𝑤) 𝑆(𝑤) 𝑇(𝑤) 𝑈(𝑤) 𝑂(𝑥,𝑦,𝑧)

**Proof of Theorem ixxssixx**
Step	Hyp	Ref	Expression
1		ixx.1	. . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
2	1	elmpocl 7511	. . 3 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
3		simp1 1135	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ^*)
4	3	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ^*))
5		simpl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
6		3simpa 1147	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤))
7		ixx.3	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤))
8	7	expimpd 454	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤) → 𝐴𝑇𝑤))
9	5, 6, 8	syl2im 40	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝐴𝑇𝑤))
10		simpr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
11		3simpb 1148	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝑤𝑆𝐵))
12		ixx.4	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))
13	12	ancoms 459	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))
14	13	expimpd 454	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → ((𝑤 ∈ ℝ^* ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵))
15	10, 11, 14	syl2im 40	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵))
16	4, 9, 15	3jcad 1128	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵)))
17	1	elixx1 13088	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
18		ixx.2	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})
19	18	elixx1 13088	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵)))
20	16, 17, 19	3imtr4d 294	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)))
21	2, 20	mpcom 38	. 2 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))
22	21	ssriv 3925	1 ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 class class class wbr 5074 (class class class)co 7275 ∈ cmpo 7277 ℝ^*cxr 11008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-xr 11013

This theorem is referenced by: ioossicc 13165 icossicc 13168 iocssicc 13169 ioossico 13170 dvloglem 25803 ioossioc 43030

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