ixxss2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ixxss2		Structured version Visualization version GIF version

Theorem ixxss2 12065

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 12059	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵)))
3	2	simplbi 475	. . . . . 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 1068	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ ℝ^*)
6	2	simprbi 479	. . . . . 6 ⊢ (𝑤 ∈ (𝐴𝑃𝐵) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
7	6	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝐴𝑅𝑤 ∧ 𝑤𝑇𝐵))
8	7	simpld 474	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴𝑅𝑤)
9	7	simprd 478	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑇𝐵)
10		simplr 788	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵𝑊𝐶)
11	4	simp2d 1067	. . . . . 6 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐵 ∈ ℝ^*)
12		simpll 786	. . . . . 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 1318	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → ((𝑤𝑇𝐵 ∧ 𝐵𝑊𝐶) → 𝑤𝑆𝐶))
15	9, 10, 14	mp2and 711	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤𝑆𝐶)
16	4	simp1d 1066	. . . . 5 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝐴 ∈ ℝ^*)
17		ixx.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
18	17	elixx1 12055	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
19	16, 12, 18	syl2anc 691	. . . 4 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
20	5, 8, 15, 19	mpbir3and 1238	. . 3 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) ∧ 𝑤 ∈ (𝐴𝑃𝐵)) → 𝑤 ∈ (𝐴𝑂𝐶))
21	20	ex 449	. 2 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝑤 ∈ (𝐴𝑃𝐵) → 𝑤 ∈ (𝐴𝑂𝐶)))
22	21	ssrdv 3574	1 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵𝑊𝐶) → (𝐴𝑃𝐵) ⊆ (𝐴𝑂𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 class class class wbr 4583 (class class class)co 6549 ↦ cmpt2 6551 ℝ^*cxr 9952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-xr 9957

This theorem is referenced by: iooss2 12082 leordtval2 20826 mnfnei 20835 psercnlem2 23982 tanord1 24087

W3C validator