elixx3g - Metamath Proof Explorer

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Theorem elixx3g 13332

Description: Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show 𝐴 ∈ ℝ^* and 𝐵 ∈ ℝ^*. (Contributed by Mario Carneiro, 3-Nov-2013.)

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

**Assertion**
Ref	Expression
elixx3g	⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 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 elixx3g**
Step	Hyp	Ref	Expression
1		anass 470	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))))
2		df-3an 1090	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^))
3	2	anbi1i 625	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)) ↔ (((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
4		ixx.1	. . . . 5 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
5	4	elixx1 13328	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
6		3anass 1096	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
7		ibar 530	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
8	6, 7	bitrid 283	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
9	5, 8	bitrd 279	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
10	4	ixxf 13329	. . . . . . 7 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11	10	fdmi 6725	. . . . . 6 ⊢ dom 𝑂 = (ℝ^* × ℝ^*)
12	11	ndmov 7585	. . . . 5 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴𝑂𝐵) = ∅)
13	12	eleq2d 2820	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ ∅))
14		noel 4328	. . . . . 6 ⊢ ¬ 𝐶 ∈ ∅
15	14	pm2.21i 119	. . . . 5 ⊢ (𝐶 ∈ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
16		simpl 484	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
17	15, 16	pm5.21ni 379	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ∅ ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
18	13, 17	bitrd 279	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
19	9, 18	pm2.61i 182	. 2 ⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))))
20	1, 3, 19	3bitr4ri 304	1 ⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {crab 3433 ∅c0 4320 𝒫 cpw 4600 class class class wbr 5146 × cxp 5672 (class class class)co 7403 ∈ cmpo 7405 ℝ^*cxr 11242

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7969 df-2nd 7970 df-xr 11247

This theorem is referenced by: ixxss1 13337 ixxss2 13338 ixxss12 13339 elioo3g 13348 elicore 13371 iccss2 13390 iccssico2 13393 xrtgioo 24303 ftc1anclem7 36504 ftc1anclem8 36505 ftc1anc 36506 eliocre 44156 lbioc 44160

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