elixx1 - Metamath Proof Explorer

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Theorem elixx1 13376

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 13375	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴𝑂𝐵) = {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)})
3	2	eleq2d 2821	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ {𝑧 ∈ ℝ^ ∣ (𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵)}))
4		breq2 5128	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝑅𝑧 ↔ 𝐴𝑅𝐶))
5		breq1 5127	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧𝑆𝐵 ↔ 𝐶𝑆𝐵))
6	4, 5	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝑧 ∧ 𝑧𝑆𝐵) ↔ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
7	6	elrab 3676	. . 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 278	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3420 class class class wbr 5124 (class class class)co 7410 ∈ cmpo 7412 ℝ^*cxr 11273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6489 df-fun 6538 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-xr 11278

This theorem is referenced by: elixx3g 13380 ixxssixx 13381 ixxdisj 13382 ixxun 13383 ixxss1 13385 ixxss2 13386 ixxss12 13387 ixxub 13388 ixxlb 13389 elioo1 13407 elioc1 13409 elico1 13410 elicc1 13411

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