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| 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 23455 | . 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) | |
| 2 | simprr 772 | . . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) = 𝐽) | |
| 3 | tgcl 22977 | . . . . 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 2841 | . . 3 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → 𝐽 ∈ Top) | 
| 6 | 5 | rexlimiva 3146 | . 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 1539 ∈ wcel 2107 ∃wrex 3069 class class class wbr 5142 ‘cfv 6560 ωcom 7888 ≼ cdom 8984 topGenctg 17483 Topctop 22900 TopBasesctb 22953 2ndωc2ndc 23447 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-topgen 17489 df-top 22901 df-bases 22954 df-2ndc 23449 | 
| This theorem is referenced by: 2ndc1stc 23460 2ndcctbss 23464 | 
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