ioomax - Metamath Proof Explorer

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Theorem ioomax 13432

Description: The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.)

**Assertion**
Ref	Expression
ioomax	⊢ (-∞(,)+∞) = ℝ

**Proof of Theorem ioomax**
Step	Hyp	Ref	Expression
1		mnfxr 11302	. . 3 ⊢ -∞ ∈ ℝ^*
2		pnfxr 11299	. . 3 ⊢ +∞ ∈ ℝ^*
3		iooval2 13390	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)})
4	1, 2, 3	mp2an 691	. 2 ⊢ (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}
5		rabid2 3461	. . 3 ⊢ (ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} ↔ ∀𝑥 ∈ ℝ (-∞ < 𝑥 ∧ 𝑥 < +∞))
6		mnflt 13136	. . . 4 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
7		ltpnf 13133	. . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
8	6, 7	jca 511	. . 3 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 ∧ 𝑥 < +∞))
9	5, 8	mprgbir 3065	. 2 ⊢ ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}
10	4, 9	eqtr4i 2759	1 ⊢ (-∞(,)+∞) = ℝ

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 class class class wbr 5148 (class class class)co 7420 ℝcr 11138 +∞cpnf 11276 -∞cmnf 11277 ℝ^*cxr 11278 < clt 11279 (,)cioo 13357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-ioo 13361

This theorem is referenced by: unirnioo 13459 resup 13865 reordt 23135 icopnfcld 24697 iocmnfcld 24698 blssioo 24724 reconnlem1 24755 ioombl1 25504 ioombl 25507 mbfdm 25568 ismbf 25570 ismbf2d 25582 ismbf3d 25596 tpr2rico 33513 esumcvgsum 33707 itgexpif 34238 retopsconn 34859 asindmre 37176 itgsubsticclem 45363

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