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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 12052 | . . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
3 | 2 | ixxex 12057 | . . . 4 ⊢ [,) ∈ V |
4 | imaexg 6995 | . . . 4 ⊢ ([,) ∈ V → ([,) “ (ℝ × ℝ)) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ([,) “ (ℝ × ℝ)) ∈ V |
6 | 1, 5 | eqeltri 2684 | . 2 ⊢ 𝐼 ∈ V |
7 | 1 | icoreclin 32381 | . . 3 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) |
8 | 7 | rgen2a 2960 | . 2 ⊢ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼 |
9 | fiinbas 20567 | . 2 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼) → 𝐼 ∈ TopBases) | |
10 | 6, 8, 9 | mp2an 704 | 1 ⊢ 𝐼 ∈ TopBases |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∩ cin 3539 × cxp 5036 “ cima 5041 ℝcr 9814 < clt 9953 ≤ cle 9954 [,)cico 12048 TopBasesctb 20520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ico 12052 df-bases 20522 |
This theorem is referenced by: istoprelowl 32384 |
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