elixx3g - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elixx3g		Structured version Visualization version GIF version

Theorem elixx3g 13295

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 1088	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ↔ ((𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ ℝ^))
3	2	anbi1i 624	. 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 13291	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴𝑅𝐶 ∧ 𝐶𝑆𝐵)))
6		3anass 1094	. . . . 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 13292	. . . . . . 7 ⊢ 𝑂:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11	10	fdmi 6681	. . . . . 6 ⊢ dom 𝑂 = (ℝ^* × ℝ^*)
12	11	ndmov 7553	. . . . 5 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴𝑂𝐵) = ∅)
13	12	eleq2d 2814	. . . 4 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴𝑂𝐵) ↔ 𝐶 ∈ ∅))
14		noel 4297	. . . . . 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 1086 = wceq 1540 ∈ wcel 2109 {crab 3402 ∅c0 4292 𝒫 cpw 4559 class class class wbr 5102 × cxp 5629 (class class class)co 7369 ∈ cmpo 7371 ℝ^*cxr 11183

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 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 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 6452 df-fun 6501 df-fn 6502 df-f 6503 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-xr 11188

This theorem is referenced by: ixxss1 13300 ixxss2 13301 ixxss12 13302 elioo3g 13311 elicore 13335 iccss2 13354 iccssico2 13357 xrtgioo 24728 ftc1anclem7 37686 ftc1anclem8 37687 ftc1anc 37688 eliocre 45500 lbioc 45504

W3C validator