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Theorem iooval2 9050

Description: Value of the open interval function. (Contributed by NM, 6-Feb-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Assertion**
Ref	Expression
iooval2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem iooval2**
Step	Hyp	Ref	Expression
1		iooval 9043	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2		inrab2 3254	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
3		ressxr 7260	. . . . . 6 ⊢ ℝ ⊆ ℝ^*
4		sseqin2 3202	. . . . . 6 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
5	3, 4	mpbi 143	. . . . 5 ⊢ (ℝ^* ∩ ℝ) = ℝ
6		rabeq 2602	. . . . 5 ⊢ ((ℝ^* ∩ ℝ) = ℝ → {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
7	5, 6	ax-mp 7	. . . 4 ⊢ {𝑥 ∈ (ℝ^* ∩ ℝ) ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
8	2, 7	eqtri 2103	. . 3 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}
9		elioore 9047	. . . . . 6 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ)
10	9	ssriv 3013	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
11	1, 10	syl6eqssr 3060	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ)
12		df-ss 2996	. . . 4 ⊢ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ⊆ ℝ ↔ ({𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
13	11, 12	sylib 120	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → ({𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ∩ ℝ) = {𝑥 ∈ ℝ^* ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
14	8, 13	syl5reqr 2130	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → {𝑥 ∈ ℝ^ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
15	1, 14	eqtrd 2115	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 {crab 2357 ∩ cin 2982 ⊆ wss 2983 class class class wbr 3806 (class class class)co 5564 ℝcr 7078 ℝ^*cxr 7250 < clt 7251 (,)cioo 9023

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-cnex 7165 ax-resscn 7166 ax-pre-ltirr 7186 ax-pre-ltwlin 7187 ax-pre-lttrn 7188

This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2612 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-id 4077 df-po 4080 df-iso 4081 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-iota 4918 df-fun 4955 df-fv 4961 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-pnf 7253 df-mnf 7254 df-xr 7255 df-ltxr 7256 df-le 7257 df-ioo 9027

This theorem is referenced by: elioo2 9056 ioomax 9083 ioopos 9085 dfioo2 9109

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