elioopnf - Metamath Proof Explorer

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Theorem elioopnf 13385

Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.)

**Assertion**
Ref	Expression
elioopnf	⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))

**Proof of Theorem elioopnf**
Step	Hyp	Ref	Expression
1		pnfxr 11188	. . 3 ⊢ +∞ ∈ ℝ^*
2		elioo2 13328	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞)))
3	1, 2	mpan2 692	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞)))
4		df-3an 1089	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞))
5		ltpnf 13060	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
6	5	adantr 480	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 < +∞)
7	6	pm4.71i 559	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ↔ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) ∧ 𝐵 < +∞))
8	4, 7	bitr4i 278	. 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
9	3, 8	bitrdi 287	1 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ (𝐴(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 𝐴 < 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7358 ℝcr 11026 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 (,)cioo 13287

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 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-pre-lttri 11101 ax-pre-lttrn 11102

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 5517 df-po 5530 df-so 5531 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-ioo 13291

This theorem is referenced by: mbfmulc2lem 25623 mbfposr 25628 ismbf3d 25630 mbfaddlem 25636 mbfsup 25640 itg2gt0 25736 itg2cnlem1 25737 itg2cnlem2 25738 lhop2 25992 dvfsumlem2 26005 dvfsumlem2OLD 26006 dvfsumlem3 26007 dvfsumrlimge0 26009 dvfsumrlim 26010 dvfsumrlim2 26011 pntpbnd1a 27567 pntpbnd2 27569 pntibndlem2 27573 pntibndlem3 27574 pntlemi 27586 pntlemo 27589 relowlssretop 37690 itg2addnclem2 38004 iblabsnclem 38015 ftc1anclem1 38025 ftc1anclem6 38030 rfcnpre1 45465 regt1loggt0 49009 rege1logbrege0 49031 rege1logbzge0 49032 io1ii 49393

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