elixx3g - Metamath Proof Explorer

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Theorem elixx3g 13311

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 468	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^) ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))))
2		df-3an 1089	. . 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 13307	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
6		3anass 1095	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵) ↔ (𝐶 ∈ ℝ^* ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
7		ibar 528	. . . . 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 13308	. . . . . . 7 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11	10	fdmi 6679	. . . . . 6 ⊢ dom 𝑂 = (ℝ^* × ℝ^*)
12	11	ndmov 7551	. . . . 5 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴𝑂𝐵) = ∅)
13	12	eleq2d 2822	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ ∅))
14		noel 4278	. . . . . 6 ⊢ ¬ 𝐶 ∈ ∅
15	14	pm2.21i 119	. . . . 5 ⊢ (𝐶 ∈ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
16		simpl 482	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ (𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
17	15, 16	pm5.21ni 377	. . . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3389 ∅c0 4273 𝒫 cpw 4541 class class class wbr 5085 × cxp 5629 (class class class)co 7367 ∈ cmpo 7369 ℝ^*cxr 11178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-xr 11183

This theorem is referenced by: ixxss1 13316 ixxss2 13317 ixxss12 13318 elioo3g 13327 elicore 13351 iccss2 13370 iccssico2 13373 xrtgioo 24772 ftc1anclem7 38020 ftc1anclem8 38021 ftc1anc 38022 eliocre 45939 lbioc 45943

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