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Theorem ixxdisj 13415

Description: Split an interval into disjoint pieces. (Contributed by Mario Carneiro, 16-Jun-2014.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxun.2	⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})
ixxun.3	⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))

**Assertion**
Ref	Expression
ixxdisj	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)

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 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 ixxdisj**
Step	Hyp	Ref	Expression
1		elin 3918	. . . 4 ⊢ (𝑤 ∈ ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) ↔ (𝑤 ∈ (𝐴𝑂𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)))
2		ixx.1	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
3	2	elixx1 13409	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
4	3	3adant3 1150	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐵) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵)))
5	4	biimpa 482	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐵))
6	5	simp3d 1162	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑤 ∈ (𝐴𝑂𝐵)) → 𝑤𝑆𝐵)
7	6	adantrr 730	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑤 ∈ (𝐴𝑂𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶))) → 𝑤𝑆𝐵)
8		ixxun.2	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑈𝑦)})
9	8	elixx1 13409	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐵𝑃𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐵𝑇𝑤 ∧ 𝑤𝑈𝐶)))
10	9	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐵𝑃𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐵𝑇𝑤 ∧ 𝑤𝑈𝐶)))
11	10	biimpa 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ ℝ^ ∧ 𝐵𝑇𝑤 ∧ 𝑤𝑈𝐶))
12	11	simp2d 1161	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤)
13		simpl2 1211	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵 ∈ ℝ^)
14	11	simp1d 1160	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ^)
15		ixxun.3	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))
16	13, 14, 15	syl2anc 596	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ↔ ¬ 𝑤𝑆𝐵))
17	12, 16	mpbid 235	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ¬ 𝑤𝑆𝐵)
18	17	adantrl 729	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑤 ∈ (𝐴𝑂𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶))) → ¬ 𝑤𝑆𝐵)
19	7, 18	pm2.65da 829	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ¬ (𝑤 ∈ (𝐴𝑂𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)))
20	19	pm2.21d 122	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑤 ∈ (𝐴𝑂𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ∅))
21	1, 20	biimtrid 245	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝑤 ∈ ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) → 𝑤 ∈ ∅))
22	21	ssrdv 3940	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) ⊆ ∅)
23		ss0 4355	. 2 ⊢ (((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) ⊆ ∅ → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)
24	22, 23	syl 18	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴𝑂𝐵) ∩ (𝐵𝑃𝐶)) = ∅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {crab 3414 ∩ cin 3901 ⊆ wss 3902 ∅c0 4282 class class class wbr 5107 (class class class)co 7416 ∈ cmpo 7418 ℝ^*cxr 11269

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-xr 11274

This theorem is used by: ioodisj 13537 lecldbas 23445 icopnfcld 24994 iocmnfcld 24995 ioombl 25794 ismbf3d 25883 joiniooico 33232 asindmre 38439 dvasin 38440 iocioodisjd 43182 iccdisj2 49810

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