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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 12374 | . . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
3 | 2 | ixxex 12379 | . . . 4 ⊢ [,) ∈ V |
4 | imaexg 7268 | . . . 4 ⊢ ([,) ∈ V → ([,) “ (ℝ × ℝ)) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ([,) “ (ℝ × ℝ)) ∈ V |
6 | 1, 5 | eqeltri 2835 | . 2 ⊢ 𝐼 ∈ V |
7 | 1 | icoreclin 33516 | . . 3 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) |
8 | 7 | rgen2a 3115 | . 2 ⊢ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼 |
9 | fiinbas 20958 | . 2 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼) → 𝐼 ∈ TopBases) | |
10 | 6, 8, 9 | mp2an 710 | 1 ⊢ 𝐼 ∈ TopBases |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 ∀wral 3050 Vcvv 3340 ∩ cin 3714 × cxp 5264 “ cima 5269 ℝcr 10127 < clt 10266 ≤ cle 10267 [,)cico 12370 TopBasesctb 20951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-pre-lttri 10202 ax-pre-lttrn 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-ico 12374 df-bases 20952 |
This theorem is referenced by: istoprelowl 33519 |
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