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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 2737 | . . 3 ⊢ (topGen‘((,) “ (ℚ × ℚ))) = (topGen‘((,) “ (ℚ × ℚ))) | |
| 2 | 1 | tgqioo 24821 | . 2 ⊢ (topGen‘ran (,)) = (topGen‘((,) “ (ℚ × ℚ))) |
| 3 | qtopbas 24780 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | |
| 4 | omelon 9686 | . . . . . 6 ⊢ ω ∈ On | |
| 5 | qnnen 16249 | . . . . . . . . 9 ⊢ ℚ ≈ ℕ | |
| 6 | xpen 9180 | . . . . . . . . 9 ⊢ ((ℚ ≈ ℕ ∧ ℚ ≈ ℕ) → (ℚ × ℚ) ≈ (ℕ × ℕ)) | |
| 7 | 5, 5, 6 | mp2an 692 | . . . . . . . 8 ⊢ (ℚ × ℚ) ≈ (ℕ × ℕ) |
| 8 | xpnnen 16247 | . . . . . . . 8 ⊢ (ℕ × ℕ) ≈ ℕ | |
| 9 | 7, 8 | entri 9048 | . . . . . . 7 ⊢ (ℚ × ℚ) ≈ ℕ |
| 10 | nnenom 14021 | . . . . . . 7 ⊢ ℕ ≈ ω | |
| 11 | 9, 10 | entr2i 9049 | . . . . . 6 ⊢ ω ≈ (ℚ × ℚ) |
| 12 | isnumi 9986 | . . . . . 6 ⊢ ((ω ∈ On ∧ ω ≈ (ℚ × ℚ)) → (ℚ × ℚ) ∈ dom card) | |
| 13 | 4, 11, 12 | mp2an 692 | . . . . 5 ⊢ (ℚ × ℚ) ∈ dom card |
| 14 | ioof 13487 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
| 15 | ffun 6739 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
| 16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ Fun (,) |
| 17 | qssre 13001 | . . . . . . . . 9 ⊢ ℚ ⊆ ℝ | |
| 18 | ressxr 11305 | . . . . . . . . 9 ⊢ ℝ ⊆ ℝ* | |
| 19 | 17, 18 | sstri 3993 | . . . . . . . 8 ⊢ ℚ ⊆ ℝ* |
| 20 | xpss12 5700 | . . . . . . . 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 4033 | . . . . . 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 10097 | . . . . 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 9048 | . . . 4 ⊢ (ℚ × ℚ) ≈ ω |
| 29 | domentr 9053 | . . . 4 ⊢ ((((,) “ (ℚ × ℚ)) ≼ (ℚ × ℚ) ∧ (ℚ × ℚ) ≈ ω) → ((,) “ (ℚ × ℚ)) ≼ ω) | |
| 30 | 27, 28, 29 | mp2an 692 | . . 3 ⊢ ((,) “ (ℚ × ℚ)) ≼ ω |
| 31 | 2ndci 23456 | . . 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 2837 | 1 ⊢ (topGen‘ran (,)) ∈ 2ndω |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 × cxp 5683 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Oncon0 6384 Fun wfun 6555 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 cardccrd 9975 ℝcr 11154 ℝ*cxr 11294 ℕcn 12266 ℚcq 12990 (,)cioo 13387 topGenctg 17482 TopBasesctb 22952 2ndωc2ndc 23446 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-topgen 17488 df-bases 22953 df-2ndc 23448 |
| This theorem is referenced by: (None) |
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