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 23408 | . . . . 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 12762 | . . . 4 ⊢ (,) ∈ V | |
9 | imaexg 7620 | . . . 4 ⊢ ((,) ∈ V → ((,) “ (ℚ × ℚ)) ∈ V) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ V |
11 | eltg3 21570 | . . 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 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 ∪ cuni 4838 × cxp 5553 ran crn 5556 “ cima 5558 ‘cfv 6355 ℚcq 12349 (,)cioo 12739 topGenctg 16711 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-ioo 12743 df-topgen 16717 df-bases 21554 |
This theorem is referenced by: smfpimbor1lem1 43093 |
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