Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgqioo2 | Structured version Visualization version GIF version |
Description: Every open set of reals is the (countable) union of open interval with rational bounds. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
tgqioo2.1 | ⊢ 𝐽 = (topGen‘ran (,)) |
tgqioo2.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
Ref | Expression |
---|---|
tgqioo2 | ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgqioo2.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐽) | |
2 | tgqioo2.1 | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | eqid 2821 | . . . . . 6 ⊢ (topGen‘((,) “ (ℚ × ℚ))) = (topGen‘((,) “ (ℚ × ℚ))) | |
4 | 3 | tgqioo 23402 | . . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘((,) “ (ℚ × ℚ))) |
5 | 2, 4, 3 | 3eqtri 2848 | . . . 4 ⊢ 𝐽 = (topGen‘((,) “ (ℚ × ℚ))) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 = (topGen‘((,) “ (ℚ × ℚ)))) |
7 | 1, 6 | eleqtrd 2915 | . 2 ⊢ (𝜑 → 𝐴 ∈ (topGen‘((,) “ (ℚ × ℚ)))) |
8 | iooex 12755 | . . . 4 ⊢ (,) ∈ V | |
9 | imaexg 7614 | . . . 4 ⊢ ((,) ∈ V → ((,) “ (ℚ × ℚ)) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ V |
11 | eltg3 21564 | . . 3 ⊢ (((,) “ (ℚ × ℚ)) ∈ V → (𝐴 ∈ (topGen‘((,) “ (ℚ × ℚ))) ↔ ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞))) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ (𝐴 ∈ (topGen‘((,) “ (ℚ × ℚ))) ↔ ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞)) |
13 | 7, 12 | sylib 220 | 1 ⊢ (𝜑 → ∃𝑞(𝑞 ⊆ ((,) “ (ℚ × ℚ)) ∧ 𝐴 = ∪ 𝑞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∪ cuni 4831 × cxp 5547 ran crn 5550 “ cima 5552 ‘cfv 6349 ℚcq 12342 (,)cioo 12732 topGenctg 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-ioo 12736 df-topgen 16711 df-bases 21548 |
This theorem is referenced by: smfpimbor1lem1 43067 |
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