Mathbox for ML |
< Previous
Next >
Nearby theorems |
||
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 12747 | . . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
3 | 2 | ixxex 12752 | . . . 4 ⊢ [,) ∈ V |
4 | imaexg 7622 | . . . 4 ⊢ ([,) ∈ V → ([,) “ (ℝ × ℝ)) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ([,) “ (ℝ × ℝ)) ∈ V |
6 | 1, 5 | eqeltri 2911 | . 2 ⊢ 𝐼 ∈ V |
7 | 1 | icoreclin 34640 | . . 3 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) |
8 | 7 | rgen2 3205 | . 2 ⊢ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼 |
9 | fiinbas 21562 | . 2 ⊢ ((𝐼 ∈ V ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼) → 𝐼 ∈ TopBases) | |
10 | 6, 8, 9 | mp2an 690 | 1 ⊢ 𝐼 ∈ TopBases |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∩ cin 3937 × cxp 5555 “ cima 5560 ℝcr 10538 < clt 10677 ≤ cle 10678 [,)cico 12743 TopBasesctb 21555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ico 12747 df-bases 21556 |
This theorem is referenced by: istoprelowl 34643 |
Copyright terms: Public domain | W3C validator |