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Theorem ixxss2 13375

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 13369	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵)))
3	2	simplbi 497	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
4	3	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
5	4	simp3d 1142	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ ℝ^*)
6	2	simprbi 496	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
7	6	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
8	7	simpld 494	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴𝑅𝑤)
9	7	simprd 495	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑇𝐵)
10		simplr 768	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵𝑊𝐶)
11	4	simp2d 1141	. . . . . 6 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵 ∈ ℝ^*)
12		simpll 766	. . . . . 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 1369	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))
15	9, 10, 14	mp2and 698	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑆𝐶)
16	4	simp1d 1140	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴 ∈ ℝ^*)
17		ixx.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
18	17	elixx1 13365	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
19	16, 12, 18	syl2anc 583	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
20	5, 8, 15, 19	mpbir3and 1340	. . 3 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ (𝐴𝑂𝐶))
21	20	ex 412	. 2 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝑤 ∈ (𝐴𝑃𝐵) → 𝑤 ∈ (𝐴𝑂𝐶)))
22	21	ssrdv 3986	1 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {crab 3429 ⊆ wss 3947 class class class wbr 5148 (class class class)co 7420 ∈ cmpo 7422 ℝ^*cxr 11277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-xr 11282

This theorem is referenced by: iooss2 13392 leordtval2 23115 mnfnei 23124 psercnlem2 26360 tanord1 26470

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