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Mirrors > Home > ILE Home > Th. List > elioomnf | GIF version |
Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
elioomnf | ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 8848 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | elioo2 8944 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴))) | |
3 | 1, 2 | mpan 414 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴))) |
4 | an32 526 | . . 3 ⊢ (((𝐵 ∈ ℝ ∧ -∞ < 𝐵) ∧ 𝐵 < 𝐴) ↔ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) ∧ -∞ < 𝐵)) | |
5 | df-3an 921 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴) ↔ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) ∧ 𝐵 < 𝐴)) | |
6 | mnflt 8858 | . . . . 5 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵) | |
7 | 6 | adantr 270 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → -∞ < 𝐵) |
8 | 7 | pm4.71i 383 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) ↔ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) ∧ -∞ < 𝐵)) |
9 | 4, 5, 8 | 3bitr4i 210 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴)) |
10 | 3, 9 | syl6bb 194 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐵 ∈ (-∞(,)𝐴) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 -∞cmnf 7151 ℝ*cxr 7152 < clt 7153 (,)cioo 8911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-ioo 8915 |
This theorem is referenced by: (None) |
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