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Theorem elioomnf 13176

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

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

**Proof of Theorem elioomnf**
Step	Hyp	Ref	Expression
1		mnfxr 11032	. . 3 ⊢ -∞ ∈ ℝ^*
2		elioo2 13120	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴)))
3	1, 2	mpan 687	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴)))
4		an32 643	. . 3 ⊢ (((𝐵 ∈ ℝ ∧ -∞ < 𝐵) ∧ 𝐵 < 𝐴) ↔ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) ∧ -∞ < 𝐵))
5		df-3an 1088	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴) ↔ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) ∧ 𝐵 < 𝐴))
6		mnflt 12859	. . . . 5 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
7	6	adantr 481	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → -∞ < 𝐵)
8	7	pm4.71i 560	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) ↔ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) ∧ -∞ < 𝐵))
9	4, 5, 8	3bitr4i 303	. 2 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴))
10	3, 9	bitrdi 287	1 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 (,)cioo 13079

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ioo 13083

This theorem is referenced by: bndth 24121 mbfmulc2lem 24811 mbfposr 24816 ismbf3d 24818 mbfi1fseqlem4 24883 itg2monolem1 24915 dvne0 25175 mbfposadd 35824 itg2addnclem2 35829 iblabsnclem 35840 ftc1anclem1 35850 ftc1anclem6 35855 rfcnpre2 42574 i0oii 46213

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