ixxss2 - Metamath Proof Explorer

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Theorem ixxss2 12511

Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxss2.2	⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)})
ixxss2.3	⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))

**Assertion**
Ref	Expression
ixxss2	⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

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

**Proof of Theorem ixxss2**
Step	Hyp	Ref	Expression
1		ixxss2.2	. . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑇𝑦)})
2	1	elixx3g 12505	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵)))
3	2	simplbi 493	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
4	3	adantl 475	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
5	4	simp3d 1135	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ ℝ^*)
6	2	simprbi 492	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
7	6	adantl 475	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
8	7	simpld 490	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴𝑅𝑤)
9	7	simprd 491	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑇𝐵)
10		simplr 759	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵𝑊𝐶)
11	4	simp2d 1134	. . . . . 6 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵 ∈ ℝ^*)
12		simpll 757	. . . . . 6 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐶 ∈ ℝ^*)
13		ixxss2.3	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))
14	5, 11, 12, 13	syl3anc 1439	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))
15	9, 10, 14	mp2and 689	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑆𝐶)
16	4	simp1d 1133	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴 ∈ ℝ^*)
17		ixx.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
18	17	elixx1 12501	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
19	16, 12, 18	syl2anc 579	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
20	5, 8, 15, 19	mpbir3and 1399	. . 3 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ (𝐴𝑂𝐶))
21	20	ex 403	. 2 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝑤 ∈ (𝐴𝑃𝐵) → 𝑤 ∈ (𝐴𝑂𝐶)))
22	21	ssrdv 3827	1 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 {crab 3094 ⊆ wss 3792 class class class wbr 4888 (class class class)co 6924 ↦ cmpt2 6926 ℝ^*cxr 10412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-xr 10417

This theorem is referenced by: iooss2 12528 leordtval2 21435 mnfnei 21444 psercnlem2 24626 tanord1 24732

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