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 12736 | . . . . 5 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
3 | 2 | ixxex 12741 | . . . 4 ⊢ [,) ∈ V |
4 | imaexg 7612 | . . . 4 ⊢ ([,) ∈ V → ([,) “ (ℝ × ℝ)) ∈ V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ([,) “ (ℝ × ℝ)) ∈ V |
6 | 1, 5 | eqeltri 2907 | . 2 ⊢ 𝐼 ∈ V |
7 | 1 | icoreclin 34630 | . . 3 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼) |
8 | 7 | rgen2 3201 | . 2 ⊢ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐼 (𝑥 ∩ 𝑦) ∈ 𝐼 |
9 | fiinbas 21552 | . 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 1531 ∈ wcel 2108 ∀wral 3136 Vcvv 3493 ∩ cin 3933 × cxp 5546 “ cima 5551 ℝcr 10528 < clt 10667 ≤ cle 10668 [,)cico 12732 TopBasesctb 21545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ico 12736 df-bases 21546 |
This theorem is referenced by: istoprelowl 34633 |
Copyright terms: Public domain | W3C validator |