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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 2795 | . . 3 ⊢ (topGen‘((,) “ (ℚ × ℚ))) = (topGen‘((,) “ (ℚ × ℚ))) | |
2 | 1 | tgqioo 23091 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘((,) “ (ℚ × ℚ))) |
3 | qtopbas 23051 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | |
4 | omelon 8955 | . . . . . 6 ⊢ ω ∈ On | |
5 | qnnen 15399 | . . . . . . . . 9 ⊢ ℚ ≈ ℕ | |
6 | xpen 8527 | . . . . . . . . 9 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ)) | |
7 | 5, 5, 6 | mp2an 688 | . . . . . . . 8 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ) |
8 | xpnnen 15397 | . . . . . . . 8 ⊢ (ℕ × ℕ) ≈ ℕ | |
9 | 7, 8 | entri 8411 | . . . . . . 7 ⊢ (ℚ × ℚ) ≈ ℕ |
10 | nnenom 13198 | . . . . . . 7 ⊢ ℕ ≈ ω | |
11 | 9, 10 | entr2i 8412 | . . . . . 6 ⊢ ω ≈ (ℚ × ℚ) |
12 | isnumi 9221 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card) | |
13 | 4, 11, 12 | mp2an 688 | . . . . 5 ⊢ (ℚ × ℚ) ∈ dom card |
14 | ioof 12685 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
15 | ffun 6385 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ Fun (,) |
17 | qssre 12208 | . . . . . . . . 9 ⊢ ℚ ⊆ ℝ | |
18 | ressxr 10531 | . . . . . . . . 9 ⊢ ℝ ⊆ ℝ* | |
19 | 17, 18 | sstri 3898 | . . . . . . . 8 ⊢ ℚ ⊆ ℝ* |
20 | xpss12 5458 | . . . . . . . 8 ⊢ ((ℚ ⊆ ℝ* ∧ ℚ ⊆ ℝ*) → (ℚ × ℚ) ⊆ (ℝ* × ℝ*)) | |
21 | 19, 19, 20 | mp2an 688 | . . . . . . 7 ⊢ (ℚ × ℚ) ⊆ (ℝ* × ℝ*) |
22 | 14 | fdmi 6392 | . . . . . . 7 ⊢ dom (,) = (ℝ* × ℝ*) |
23 | 21, 22 | sseqtr4i 3925 | . . . . . 6 ⊢ (ℚ × ℚ) ⊆ dom (,) |
24 | fores 6468 | . . . . . 6 ⊢ ((Fun (,) ∧ (ℚ × ℚ) ⊆ dom (,)) → ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ))) | |
25 | 16, 23, 24 | mp2an 688 | . . . . 5 ⊢ ((,) ↾ (ℚ × ℚ)):(ℚ × ℚ)–onto→((,) “ (ℚ × ℚ)) |
26 | fodomnum 9329 | . . . . 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 8411 | . . . 4 ⊢ (ℚ × ℚ) ≈ ω |
29 | domentr 8416 | . . . 4 ⊢ ((((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ω) → ((,) “ (ℚ × ℚ)) ≼ ω) | |
30 | 27, 28, 29 | mp2an 688 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω |
31 | 2ndci 21740 | . . 3 ⊢ ((((,) “ (ℚ × ℚ)) ∈ TopBases ∧ ((,) “ (ℚ × ℚ)) ≼ ω) → (topGen‘((,) “ (ℚ × ℚ))) ∈ 2nd𝜔) | |
32 | 3, 30, 31 | mp2an 688 | . 2 ⊢ (topGen‘((,) “ (ℚ × ℚ))) ∈ 2nd𝜔 |
33 | 2, 32 | eqeltri 2879 | 1 ⊢ (topGen‘ran (,)) ∈ 2nd𝜔 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2081 ⊆ wss 3859 𝒫 cpw 4453 class class class wbr 4962 × cxp 5441 dom cdm 5443 ran crn 5444 ↾ cres 5445 “ cima 5446 Oncon0 6066 Fun wfun 6219 ⟶wf 6221 –onto→wfo 6223 ‘cfv 6225 ωcom 7436 ≈ cen 8354 ≼ cdom 8355 cardccrd 9210 ℝcr 10382 ℝ*cxr 10520 ℕcn 11486 ℚcq 12197 (,)cioo 12588 topGenctg 16540 TopBasesctb 21237 2nd𝜔c2ndc 21730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-omul 7958 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-acn 9217 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-ioo 12592 df-topgen 16546 df-bases 21238 df-2ndc 21732 |
This theorem is referenced by: (None) |
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