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Mirrors > Home > MPE Home > Th. List > re2ndc | Structured version Visualization version GIF version |
Description: The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
re2ndc | ⊢ (topGen‘ran (,)) ∈ 2ndω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . 3 ⊢ (topGen‘((,) “ (ℚ × ℚ))) = (topGen‘((,) “ (ℚ × ℚ))) | |
2 | 1 | tgqioo 23411 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘((,) “ (ℚ × ℚ))) |
3 | qtopbas 23371 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | |
4 | omelon 9112 | . . . . . 6 ⊢ ω ∈ On | |
5 | qnnen 15569 | . . . . . . . . 9 ⊢ ℚ ≈ ℕ | |
6 | xpen 8683 | . . . . . . . . 9 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ)) | |
7 | 5, 5, 6 | mp2an 690 | . . . . . . . 8 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ) |
8 | xpnnen 15567 | . . . . . . . 8 ⊢ (ℕ × ℕ) ≈ ℕ | |
9 | 7, 8 | entri 8566 | . . . . . . 7 ⊢ (ℚ × ℚ) ≈ ℕ |
10 | nnenom 13351 | . . . . . . 7 ⊢ ℕ ≈ ω | |
11 | 9, 10 | entr2i 8567 | . . . . . 6 ⊢ ω ≈ (ℚ × ℚ) |
12 | isnumi 9378 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card) | |
13 | 4, 11, 12 | mp2an 690 | . . . . 5 ⊢ (ℚ × ℚ) ∈ dom card |
14 | ioof 12838 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
15 | ffun 6520 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ Fun (,) |
17 | qssre 12361 | . . . . . . . . 9 ⊢ ℚ ⊆ ℝ | |
18 | ressxr 10688 | . . . . . . . . 9 ⊢ ℝ ⊆ ℝ* | |
19 | 17, 18 | sstri 3979 | . . . . . . . 8 ⊢ ℚ ⊆ ℝ* |
20 | xpss12 5573 | . . . . . . . 8 ⊢ ((ℚ ⊆ ℝ* ∧ ℚ ⊆ ℝ*) → (ℚ × ℚ) ⊆ (ℝ* × ℝ*)) | |
21 | 19, 19, 20 | mp2an 690 | . . . . . . 7 ⊢ (ℚ × ℚ) ⊆ (ℝ* × ℝ*) |
22 | 14 | fdmi 6527 | . . . . . . 7 ⊢ dom (,) = (ℝ* × ℝ*) |
23 | 21, 22 | sseqtrri 4007 | . . . . . 6 ⊢ (ℚ × ℚ) ⊆ dom (,) |
24 | fores 6603 | . . . . . 6 ⊢ ((Fun (,) ∧ (ℚ × ℚ) ⊆ dom (,)) → ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ))) | |
25 | 16, 23, 24 | mp2an 690 | . . . . 5 ⊢ ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)) |
26 | fodomnum 9486 | . . . . 5 ⊢ ((ℚ × ℚ) ∈ dom card → (((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)) → ((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ))) | |
27 | 13, 25, 26 | mp2 9 | . . . 4 ⊢ ((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ) |
28 | 9, 10 | entri 8566 | . . . 4 ⊢ (ℚ × ℚ) ≈ ω |
29 | domentr 8571 | . . . 4 ⊢ ((((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ω) → ((,) “ (ℚ × ℚ)) ≼ ω) | |
30 | 27, 28, 29 | mp2an 690 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω |
31 | 2ndci 22059 | . . 3 ⊢ ((((,) “ (ℚ × ℚ)) ∈ TopBases ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → (topGen‘((,) “ (ℚ × ℚ))) ∈ 2ndω) | |
32 | 3, 30, 31 | mp2an 690 | . 2 ⊢ (topGen‘((,) “ (ℚ × ℚ))) ∈ 2ndω |
33 | 2, 32 | eqeltri 2912 | 1 ⊢ (topGen‘ran (,)) ∈ 2ndω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2113 ⊆ wss 3939 𝒫 cpw 4542 class class class wbr 5069 × cxp 5556 dom cdm 5558 ran crn 5559 ↾ cres 5560 “ cima 5561 Oncon0 6194 Fun wfun 6352 ⟶wf 6354 –onto→wfo 6356 ‘cfv 6358 ωcom 7583 ≈ cen 8509 ≼ cdom 8510 cardccrd 9367 ℝcr 10539 ℝ*cxr 10677 ℕcn 11641 ℚcq 12351 (,)cioo 12741 topGenctg 16714 TopBasesctb 21556 2ndωc2ndc 22049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-acn 9374 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-ioo 12745 df-topgen 16720 df-bases 21557 df-2ndc 22051 |
This theorem is referenced by: (None) |
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