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Theorem ixxssixx 9977

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
ixxssixx.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,𝑥 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(𝑦,𝑧)

**Proof of Theorem ixxssixx**
Step	Hyp	Ref	Expression
1		ixxssixx.1	. . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
2	1	elmpocl 6118	. . 3 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
3		simp1 999	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ^*)
4	3	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤 ∈ ℝ^*))
5		simpl 109	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
6		3simpa 996	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤))
7		ixx.3	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴𝑅𝑤 → 𝐴𝑇𝑤))
8	7	expimpd 363	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤) → 𝐴𝑇𝑤))
9	5, 6, 8	syl2im 38	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝐴𝑇𝑤))
10		simpr 110	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐵 ∈ ℝ^)
11		3simpb 997	. . . . . 6 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝑤𝑆𝐵))
12		ixx.4	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))
13	12	ancoms 268	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝑤𝑆𝐵 → 𝑤𝑈𝐵))
14	13	expimpd 363	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → ((𝑤 ∈ ℝ^* ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵))
15	10, 11, 14	syl2im 38	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → 𝑤𝑈𝐵))
16	4, 9, 15	3jcad 1180	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵) → (𝑤 ∈ ℝ^* ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵)))
17	1	elixx1 9972	. . . 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 9972	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑃𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑇𝑤 ∧ 𝑤𝑈𝐵)))
20	16, 17, 19	3imtr4d 203	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵)))
21	2, 20	mpcom 36	. 2 ⊢ (𝑤 ∈ (𝐴𝑂𝐵) → 𝑤 ∈ (𝐴𝑃𝐵))
22	21	ssriv 3187	1 ⊢ (𝐴𝑂𝐵) ⊆ (𝐴𝑃𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 {crab 2479 ⊆ wss 3157 class class class wbr 4033 (class class class)co 5922 ∈ cmpo 5924 ℝ^*cxr 8060

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-xr 8065

This theorem is referenced by: ioossicc 10034 icossicc 10035 iocssicc 10036 ioossico 10037

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