ioopos - Metamath Proof Explorer

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Theorem ioopos 12891

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 10759	. . 3 ⊢ 0 ∈ ℝ^*
2		pnfxr 10766	. . 3 ⊢ +∞ ∈ ℝ^*
3		iooval2 12847	. . 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 12591	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
6	5	biantrud 535	. . 3 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ (0 < 𝑥 ∧ 𝑥 < +∞)))
7	6	rabbiia 3372	. 2 ⊢ {𝑥 ∈ ℝ ∣ 0 < 𝑥} = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)}
8	4, 7	eqtr4i 2764	1 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥}

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 = wceq 1542 ∈ wcel 2113 {crab 3057 class class class wbr 5027 (class class class)co 7164 ℝcr 10607 0cc0 10608 +∞cpnf 10743 ℝ^*cxr 10745 < clt 10746 (,)cioo 12814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-addrcl 10669 ax-rnegex 10679 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-ioo 12818

This theorem is referenced by: ioorp 12892 repos 12913

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