unirnioo - Metamath Proof Explorer

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Theorem unirnioo 13349

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 13322	. . . 4 ⊢ (-∞(,)+∞) = ℝ
2		ioof 13347	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6651	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		mnfxr 11169	. . . . 5 ⊢ -∞ ∈ ℝ^*
6		pnfxr 11166	. . . . 5 ⊢ +∞ ∈ ℝ^*
7		fnovrn 7521	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
8	4, 5, 6, 7	mp3an 1463	. . . 4 ⊢ (-∞(,)+∞) ∈ ran (,)
9	1, 8	eqeltrri 2828	. . 3 ⊢ ℝ ∈ ran (,)
10		elssuni 4887	. . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,))
11	9, 10	ax-mp 5	. 2 ⊢ ℝ ⊆ ∪ ran (,)
12		frn 6658	. . . 4 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
13	2, 12	ax-mp 5	. . 3 ⊢ ran (,) ⊆ 𝒫 ℝ
14		sspwuni 5046	. . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ)
15	13, 14	mpbi 230	. 2 ⊢ ∪ ran (,) ⊆ ℝ
16	11, 15	eqssi 3946	1 ⊢ ℝ = ∪ ran (,)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 ⊆ wss 3897 𝒫 cpw 4547 ∪ cuni 4856 × cxp 5612 ran crn 5615 Fn wfn 6476 ⟶wf 6477 (class class class)co 7346 ℝcr 11005 +∞cpnf 11143 -∞cmnf 11144 ℝ^*cxr 11145 (,)cioo 13245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-ioo 13249

This theorem is referenced by: pnfnei 23135 mnfnei 23136 uniretop 24677 tgioo 24711 xrtgioo 24722 bndth 24884 relowlssretop 37407 relowlpssretop 37408 mblfinlem3 37709 mblfinlem4 37710 ismblfin 37711

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