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Theorem elioomnf 13341

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 11166	. . 3 ⊢ -∞ ∈ ℝ^*
2		elioo2 13283	. . 3 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴)))
3	1, 2	mpan 690	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴)))
4		an32 646	. . 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 13019	. . . . 5 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
7	6	adantr 480	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → -∞ < 𝐵)
8	7	pm4.71i 559	. . 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 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2111 class class class wbr 5091 (class class class)co 7346 ℝcr 11002 -∞cmnf 11141 ℝ^*cxr 11142 < clt 11143 (,)cioo 13242

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 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-pre-lttri 11077 ax-pre-lttrn 11078

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 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 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 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-ioo 13246

This theorem is referenced by: bndth 24882 mbfmulc2lem 25573 mbfposr 25578 ismbf3d 25580 mbfi1fseqlem4 25644 itg2monolem1 25676 dvne0 25941 mbfposadd 37706 itg2addnclem2 37711 iblabsnclem 37722 ftc1anclem1 37732 ftc1anclem6 37737 redvmptabs 42392 rfcnpre2 45067 i0oii 48950

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