elixx3g - Metamath Proof Explorer

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Theorem elixx3g 13513

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 474	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))))
2		df-3an 1105	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^))
3	2	anbi1i 636	. 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 13509	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
6		3anass 1111	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
7		ibar 538	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
8	6, 7	bitrid 286	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
9	5, 8	bitrd 282	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
10	4	ixxf 13510	. . . . . . 7 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11	10	fdmi 6740	. . . . . 6 ⊢ dom 𝑂 = (ℝ^* × ℝ^*)
12	11	ndmov 7626	. . . . 5 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴𝑂𝐵) = ∅)
13	12	eleq2d 2850	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ ∅))
14		noel 4286	. . . . . 6 ⊢ ¬ 𝐶 ∈ ∅
15	14	pm2.21i 120	. . . . 5 ⊢ (𝐶 ∈ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
16		simpl 488	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
17	15, 16	pm5.21ni 380	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ ∅ ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
18	13, 17	bitrd 282	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))))
19	9, 18	pm2.61i 184	. 2 ⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))))
20	1, 3, 19	3bitr4ri 307	1 ⊢ (𝐶 ∈ (𝐴𝑂𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {crab 3416 ∅c0 4281 𝒫 cpw 4560 class class class wbr 5106 × cxp 5653 (class class class)co 7440 ∈ cmpo 7442 ℝ^*cxr 11366

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-fv 6563 df-ov 7443 df-oprab 7444 df-mpo 7445 df-1st 8022 df-2nd 8023 df-xr 11371

This theorem is used by: ixxss1 13518 ixxss2 13519 ixxss12 13520 elioo3g 13529 elicore 13553 iccss2 13572 iccssico2 13575 xrtgioo 25153 ftc1anclem7 38808 ftc1anclem8 38809 ftc1anc 38810 eliocre 46724 lbioc 46728

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