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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 2734 | . . 3 ⊢ (topGen‘((,) “ (ℚ × ℚ))) = (topGen‘((,) “ (ℚ × ℚ))) | |
2 | 1 | tgqioo 24835 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘((,) “ (ℚ × ℚ))) |
3 | qtopbas 24795 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | |
4 | omelon 9683 | . . . . . 6 ⊢ ω ∈ On | |
5 | qnnen 16245 | . . . . . . . . 9 ⊢ ℚ ≈ ℕ | |
6 | xpen 9178 | . . . . . . . . 9 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ)) | |
7 | 5, 5, 6 | mp2an 692 | . . . . . . . 8 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ) |
8 | xpnnen 16243 | . . . . . . . 8 ⊢ (ℕ × ℕ) ≈ ℕ | |
9 | 7, 8 | entri 9046 | . . . . . . 7 ⊢ (ℚ × ℚ) ≈ ℕ |
10 | nnenom 14017 | . . . . . . 7 ⊢ ℕ ≈ ω | |
11 | 9, 10 | entr2i 9047 | . . . . . 6 ⊢ ω ≈ (ℚ × ℚ) |
12 | isnumi 9983 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card) | |
13 | 4, 11, 12 | mp2an 692 | . . . . 5 ⊢ (ℚ × ℚ) ∈ dom card |
14 | ioof 13483 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
15 | ffun 6739 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ Fun (,) |
17 | qssre 12998 | . . . . . . . . 9 ⊢ ℚ ⊆ ℝ | |
18 | ressxr 11302 | . . . . . . . . 9 ⊢ ℝ ⊆ ℝ* | |
19 | 17, 18 | sstri 4004 | . . . . . . . 8 ⊢ ℚ ⊆ ℝ* |
20 | xpss12 5703 | . . . . . . . 8 ⊢ ((ℚ ⊆ ℝ* ∧ ℚ ⊆ ℝ*) → (ℚ × ℚ) ⊆ (ℝ* × ℝ*)) | |
21 | 19, 19, 20 | mp2an 692 | . . . . . . 7 ⊢ (ℚ × ℚ) ⊆ (ℝ* × ℝ*) |
22 | 14 | fdmi 6747 | . . . . . . 7 ⊢ dom (,) = (ℝ* × ℝ*) |
23 | 21, 22 | sseqtrri 4032 | . . . . . 6 ⊢ (ℚ × ℚ) ⊆ dom (,) |
24 | fores 6830 | . . . . . 6 ⊢ ((Fun (,) ∧ (ℚ × ℚ) ⊆ dom (,)) → ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ))) | |
25 | 16, 23, 24 | mp2an 692 | . . . . 5 ⊢ ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)) |
26 | fodomnum 10094 | . . . . 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 9046 | . . . 4 ⊢ (ℚ × ℚ) ≈ ω |
29 | domentr 9051 | . . . 4 ⊢ ((((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ω) → ((,) “ (ℚ × ℚ)) ≼ ω) | |
30 | 27, 28, 29 | mp2an 692 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω |
31 | 2ndci 23471 | . . 3 ⊢ ((((,) “ (ℚ × ℚ)) ∈ TopBases ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → (topGen‘((,) “ (ℚ × ℚ))) ∈ 2ndω) | |
32 | 3, 30, 31 | mp2an 692 | . 2 ⊢ (topGen‘((,) “ (ℚ × ℚ))) ∈ 2ndω |
33 | 2, 32 | eqeltri 2834 | 1 ⊢ (topGen‘ran (,)) ∈ 2ndω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⊆ wss 3962 𝒫 cpw 4604 class class class wbr 5147 × cxp 5686 dom cdm 5688 ran crn 5689 ↾ cres 5690 “ cima 5691 Oncon0 6385 Fun wfun 6556 ⟶wf 6558 –onto→wfo 6560 ‘cfv 6562 ωcom 7886 ≈ cen 8980 ≼ cdom 8981 cardccrd 9972 ℝcr 11151 ℝ*cxr 11291 ℕcn 12263 ℚcq 12987 (,)cioo 13383 topGenctg 17483 TopBasesctb 22967 2ndωc2ndc 23461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-ioo 13387 df-topgen 17489 df-bases 22968 df-2ndc 23463 |
This theorem is referenced by: (None) |
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