ixxssixx - Metamath Proof Explorer

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Theorem ixxssixx 12402

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	elmpt2cl 7042	. . 3 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
3		simp1 1131	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ^*)
4	3	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ^*))
5		simpl 474	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
6		3simpa 1143	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤))
7		ixx.3	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤))
8	7	expimpd 630	. . . . . 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 479	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
11		3simpb 1145	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝑤𝑆𝐵))
12		ixx.4	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))
13	12	ancoms 468	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))
14	13	expimpd 630	. . . . . 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 1124	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵)))
17	1	elixx1 12397	. . . 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 12397	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵)))
20	16, 17, 19	3imtr4d 283	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)))
21	2, 20	mpcom 38	. 2 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))
22	21	ssriv 3748	1 ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 {crab 3054 ⊆ wss 3715 class class class wbr 4804 (class class class)co 6814 ↦ cmpt2 6816 ℝ^*cxr 10285

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 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-xr 10290

This theorem is referenced by: ioossicc 12472 icossicc 12473 iocssicc 12474 ioossico 12475 dvloglem 24614 ioossioc 40234

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