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Mirrors > Home > MPE Home > Th. List > 2ndctop | Structured version Visualization version GIF version |
Description: A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
2ndctop | ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | is2ndc 23470 | . 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) | |
2 | simprr 773 | . . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) = 𝐽) | |
3 | tgcl 22992 | . . . . 5 ⊢ (𝑥 ∈ TopBases → (topGen‘𝑥) ∈ Top) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) ∈ Top) |
5 | 2, 4 | eqeltrrd 2840 | . . 3 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → 𝐽 ∈ Top) |
6 | 5 | rexlimiva 3145 | . 2 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ Top) |
7 | 1, 6 | sylbi 217 | 1 ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 ωcom 7887 ≼ cdom 8982 topGenctg 17484 Topctop 22915 TopBasesctb 22968 2ndωc2ndc 23462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-topgen 17490 df-top 22916 df-bases 22969 df-2ndc 23464 |
This theorem is referenced by: 2ndc1stc 23475 2ndcctbss 23479 |
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