unirnioo - Metamath Proof Explorer

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Theorem unirnioo 13397

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 13370	. . . 4 ⊢ (-∞(,)+∞) = ℝ
2		ioof 13395	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6664	. . . . . 6 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		mnfxr 11197	. . . . 5 ⊢ -∞ ∈ ℝ^*
6		pnfxr 11194	. . . . 5 ⊢ +∞ ∈ ℝ^*
7		fnovrn 7537	. . . . 5 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) ∈ ran (,))
8	4, 5, 6, 7	mp3an 1464	. . . 4 ⊢ (-∞(,)+∞) ∈ ran (,)
9	1, 8	eqeltrri 2834	. . 3 ⊢ ℝ ∈ ran (,)
10		elssuni 4882	. . 3 ⊢ (ℝ ∈ ran (,) → ℝ ⊆ ∪ ran (,))
11	9, 10	ax-mp 5	. 2 ⊢ ℝ ⊆ ∪ ran (,)
12		frn 6671	. . . 4 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
13	2, 12	ax-mp 5	. . 3 ⊢ ran (,) ⊆ 𝒫 ℝ
14		sspwuni 5043	. . 3 ⊢ (ran (,) ⊆ 𝒫 ℝ ↔ ∪ ran (,) ⊆ ℝ)
15	13, 14	mpbi 230	. 2 ⊢ ∪ ran (,) ⊆ ℝ
16	11, 15	eqssi 3939	1 ⊢ ℝ = ∪ ran (,)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 ⊆ wss 3890 𝒫 cpw 4542 ∪ cuni 4851 × cxp 5624 ran crn 5627 Fn wfn 6489 ⟶wf 6490 (class class class)co 7362 ℝcr 11032 +∞cpnf 11171 -∞cmnf 11172 ℝ^*cxr 11173 (,)cioo 13293

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-pre-lttri 11107 ax-pre-lttrn 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5521 df-po 5534 df-so 5535 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7937 df-2nd 7938 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-ioo 13297

This theorem is referenced by: pnfnei 23199 mnfnei 23200 uniretop 24741 tgioo 24775 xrtgioo 24786 bndth 24939 relowlssretop 37697 relowlpssretop 37698 mblfinlem3 37998 mblfinlem4 37999 ismblfin 38000

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