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Theorem 2ndctop 21472
Description: A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndctop (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ Top)

Proof of Theorem 2ndctop
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 is2ndc 21471 . 2 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2 simprr 813 . . . 4 ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) = 𝐽)
3 tgcl 20995 . . . . 5 (𝑥 ∈ TopBases → (topGen‘𝑥) ∈ Top)
43adantr 472 . . . 4 ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) ∈ Top)
52, 4eqeltrrd 2840 . . 3 ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → 𝐽 ∈ Top)
65rexlimiva 3166 . 2 (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ Top)
71, 6sylbi 207 1 (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051   class class class wbr 4804  cfv 6049  ωcom 7231  cdom 8121  topGenctg 16320  Topctop 20920  TopBasesctb 20971  2nd𝜔c2ndc 21463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-topgen 16326  df-top 20921  df-bases 20972  df-2ndc 21465
This theorem is referenced by:  2ndc1stc  21476  2ndcctbss  21480
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