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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 2725 | . . 3 ⊢ (topGen‘((,) “ (ℚ × ℚ))) = (topGen‘((,) “ (ℚ × ℚ))) | |
2 | 1 | tgqioo 24765 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘((,) “ (ℚ × ℚ))) |
3 | qtopbas 24725 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | |
4 | omelon 9676 | . . . . . 6 ⊢ ω ∈ On | |
5 | qnnen 16198 | . . . . . . . . 9 ⊢ ℚ ≈ ℕ | |
6 | xpen 9168 | . . . . . . . . 9 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ)) | |
7 | 5, 5, 6 | mp2an 690 | . . . . . . . 8 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ) |
8 | xpnnen 16196 | . . . . . . . 8 ⊢ (ℕ × ℕ) ≈ ℕ | |
9 | 7, 8 | entri 9029 | . . . . . . 7 ⊢ (ℚ × ℚ) ≈ ℕ |
10 | nnenom 13986 | . . . . . . 7 ⊢ ℕ ≈ ω | |
11 | 9, 10 | entr2i 9030 | . . . . . 6 ⊢ ω ≈ (ℚ × ℚ) |
12 | isnumi 9976 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card) | |
13 | 4, 11, 12 | mp2an 690 | . . . . 5 ⊢ (ℚ × ℚ) ∈ dom card |
14 | ioof 13464 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
15 | ffun 6726 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ Fun (,) |
17 | qssre 12981 | . . . . . . . . 9 ⊢ ℚ ⊆ ℝ | |
18 | ressxr 11295 | . . . . . . . . 9 ⊢ ℝ ⊆ ℝ* | |
19 | 17, 18 | sstri 3986 | . . . . . . . 8 ⊢ ℚ ⊆ ℝ* |
20 | xpss12 5693 | . . . . . . . 8 ⊢ ((ℚ ⊆ ℝ* ∧ ℚ ⊆ ℝ*) → (ℚ × ℚ) ⊆ (ℝ* × ℝ*)) | |
21 | 19, 19, 20 | mp2an 690 | . . . . . . 7 ⊢ (ℚ × ℚ) ⊆ (ℝ* × ℝ*) |
22 | 14 | fdmi 6734 | . . . . . . 7 ⊢ dom (,) = (ℝ* × ℝ*) |
23 | 21, 22 | sseqtrri 4014 | . . . . . 6 ⊢ (ℚ × ℚ) ⊆ dom (,) |
24 | fores 6820 | . . . . . 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 10087 | . . . . 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 9029 | . . . 4 ⊢ (ℚ × ℚ) ≈ ω |
29 | domentr 9034 | . . . 4 ⊢ ((((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ω) → ((,) “ (ℚ × ℚ)) ≼ ω) | |
30 | 27, 28, 29 | mp2an 690 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω |
31 | 2ndci 23401 | . . 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 2821 | 1 ⊢ (topGen‘ran (,)) ∈ 2ndω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ⊆ wss 3944 𝒫 cpw 4604 class class class wbr 5149 × cxp 5676 dom cdm 5678 ran crn 5679 ↾ cres 5680 “ cima 5681 Oncon0 6371 Fun wfun 6543 ⟶wf 6545 –onto→wfo 6547 ‘cfv 6549 ωcom 7871 ≈ cen 8961 ≼ cdom 8962 cardccrd 9965 ℝcr 11144 ℝ*cxr 11284 ℕcn 12250 ℚcq 12970 (,)cioo 13364 topGenctg 17427 TopBasesctb 22897 2ndωc2ndc 23391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-acn 9972 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-ioo 13368 df-topgen 17433 df-bases 22898 df-2ndc 23393 |
This theorem is referenced by: (None) |
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