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Mirrors > Home > ILE Home > Th. List > ubioog | GIF version |
Description: An open interval does not contain its right endpoint. (Contributed by Jim Kingdon, 30-Mar-2020.) |
Ref | Expression |
---|---|
ubioog | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴(,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltnr 9692 | . . . 4 ⊢ (𝐵 ∈ ℝ* → ¬ 𝐵 < 𝐵) | |
2 | simp3 984 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 < 𝐵) → 𝐵 < 𝐵) | |
3 | 1, 2 | nsyl 618 | . . 3 ⊢ (𝐵 ∈ ℝ* → ¬ (𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 < 𝐵)) |
4 | 3 | adantl 275 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ (𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 < 𝐵)) |
5 | elioo1 9821 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 ∈ (𝐴(,)𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ∧ 𝐵 < 𝐵))) | |
6 | 4, 5 | mtbird 663 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ¬ 𝐵 ∈ (𝐴(,)𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 2128 class class class wbr 3967 (class class class)co 5826 ℝ*cxr 7913 < clt 7914 (,)cioo 9798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-pre-ltirr 7846 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-ioo 9802 |
This theorem is referenced by: (None) |
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