ioopos - Metamath Proof Explorer

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Theorem ioopos 13441

Description: The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.)

**Assertion**
Ref	Expression
ioopos	⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}

**Proof of Theorem ioopos**
Step	Hyp	Ref	Expression
1		0xr 11282	. . 3 ⊢ 0 ∈ ℝ^*
2		pnfxr 11289	. . 3 ⊢ +∞ ∈ ℝ^*
3		iooval2 13395	. . 3 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (0(,)+∞) = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)})
4	1, 2, 3	mp2an 692	. 2 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)}
5		ltpnf 13136	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
6	5	biantrud 531	. . 3 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ (0 < 𝑥 ∧ 𝑥 < +∞)))
7	6	rabbiia 3419	. 2 ⊢ {𝑥 ∈ ℝ ∣ 0 < 𝑥} = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)}
8	4, 7	eqtr4i 2761	1 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3415 class class class wbr 5119 (class class class)co 7405 ℝcr 11128 0cc0 11129 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 (,)cioo 13362

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-addrcl 11190 ax-rnegex 11200 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7988 df-2nd 7989 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-ioo 13366

This theorem is referenced by: ioorp 13442 repos 13463

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