unirnioo - Metamath Proof Explorer

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Theorem unirnioo 13352

Description: The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

**Assertion**
Ref	Expression
unirnioo	⊢ ℝ = ∪ ran (,)

**Proof of Theorem unirnioo**
Step	Hyp	Ref	Expression
1		ioomax 13325	. . . 4 ⊢ (-∞(,)+∞) = ℝ
2		ioof 13350	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6652	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		mnfxr 11172	. . . . 5 ⊢ -∞ ∈ ℝ^*
6		pnfxr 11169	. . . . 5 ⊢ +∞ ∈ ℝ^*
7		fnovrn 7524	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
8	4, 5, 6, 7	mp3an 1463	. . . 4 ⊢ (-∞(,)+∞) ∈ ran (,)
9	1, 8	eqeltrri 2825	. . 3 ⊢ ℝ ∈ ran (,)
10		elssuni 4888	. . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,))
11	9, 10	ax-mp 5	. 2 ⊢ ℝ ⊆ ∪ ran (,)
12		frn 6659	. . . 4 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
13	2, 12	ax-mp 5	. . 3 ⊢ ran (,) ⊆ 𝒫 ℝ
14		sspwuni 5049	. . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ)
15	13, 14	mpbi 230	. 2 ⊢ ∪ ran (,) ⊆ ℝ
16	11, 15	eqssi 3952	1 ⊢ ℝ = ∪ ran (,)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ⊆ wss 3903 𝒫 cpw 4551 ∪ cuni 4858 × cxp 5617 ran crn 5620 Fn wfn 6477 ⟶wf 6478 (class class class)co 7349 ℝcr 11008 +∞cpnf 11146 -∞cmnf 11147 ℝ^*cxr 11148 (,)cioo 13248

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 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-po 5527 df-so 5528 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-ioo 13252

This theorem is referenced by: pnfnei 23105 mnfnei 23106 uniretop 24648 tgioo 24682 xrtgioo 24693 bndth 24855 relowlssretop 37347 relowlpssretop 37348 mblfinlem3 37649 mblfinlem4 37650 ismblfin 37651

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