elixx1 - Metamath Proof Explorer

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Theorem elixx1 13088

Description: Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

**Hypothesis**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})

**Assertion**
Ref	Expression
elixx1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))

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

**Proof of Theorem elixx1**
Step	Hyp	Ref	Expression
1		ixx.1	. . . 4 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
2	1	ixxval 13087	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
3	2	eleq2d 2824	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)}))
4		breq2 5078	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝑅𝑧 ↔ 𝐴𝑅𝐶))
5		breq1 5077	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧𝑆𝐵 ↔ 𝐶𝑆𝐵))
6	4, 5	anbi12d 631	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ↔ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
7	6	elrab 3624	. . 3 ⊢ (𝐶 ∈ {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ↔ (𝐶 ∈ ℝ^* ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
8		3anass 1094	. . 3 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
9	7, 8	bitr4i 277	. 2 ⊢ (𝐶 ∈ {𝑧 ∈ ℝ^* ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)} ↔ (𝐶 ∈ ℝ^* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))
10	3, 9	bitrdi 287	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {crab 3068 class class class wbr 5074 (class class class)co 7275 ∈ cmpo 7277 ℝ^*cxr 11008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-xr 11013

This theorem is referenced by: elixx3g 13092 ixxssixx 13093 ixxdisj 13094 ixxun 13095 ixxss1 13097 ixxss2 13098 ixxss12 13099 ixxub 13100 ixxlb 13101 elioo1 13119 elioc1 13121 elico1 13122 elicc1 13123

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