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Mirrors > Home > MPE Home > Th. List > Mathboxes > isbasisrelowl | Structured version Visualization version GIF version |
Description: The set of all closed-below, open-above intervals of reals form a basis. (Contributed by ML, 27-Jul-2020.) |
Ref | Expression |
---|---|
isbasisrelowl.1 | ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) |
Ref | Expression |
---|---|
isbasisrelowl | ⊢ 𝐼 ∈ TopBases |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isbasisrelowl.1 | . . 3 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
2 | df-ico 12732 | . . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
3 | 2 | ixxex 12737 | . . . 4 ⊢ [,) ∈ V |
4 | imaexg 7602 | . . . 4 ⊢ ([,) ∈ V → ([,) “ (ℝ × ℝ)) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ([,) “ (ℝ × ℝ)) ∈ V |
6 | 1, 5 | eqeltri 2886 | . 2 ⊢ 𝐼 ∈ V |
7 | 1 | icoreclin 34774 | . . 3 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) |
8 | 7 | rgen2 3168 | . 2 ⊢ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼 |
9 | fiinbas 21557 | . 2 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼) → 𝐼 ∈ TopBases) | |
10 | 6, 8, 9 | mp2an 691 | 1 ⊢ 𝐼 ∈ TopBases |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∩ cin 3880 × cxp 5517 “ cima 5522 ℝcr 10525 < clt 10664 ≤ cle 10665 [,)cico 12728 TopBasesctb 21550 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 df-bases 21551 |
This theorem is referenced by: istoprelowl 34777 |
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