Theorem 2ndctop 23772

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.

Step	Hyp	Ref	Expression
1		is2ndc 23771	. 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2		simprr 785	. . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) = 𝐽)
3		tgcl 23294	. . . . 5 ⊢ (𝑥 ∈ TopBases → (topGen‘𝑥) ∈ Top)
4	3	adantr 486	. . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) ∈ Top)
5	2, 4	eqeltrrd 2861	. . 3 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → 𝐽 ∈ Top)
6	5	rexlimiva 3155	. 2 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ Top)
7	1, 6	sylbi 220	1 ⊢ (𝐽 ∈ 2ndω → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ‘cfv 6541  ωcom 7879   ≼ cdom 8976  topGenctg 17615  Topctop 23218  TopBasesctb 23270  2^ndωc2ndc 23763

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6496  df-fun 6543  df-fv 6549  df-topgen 17621  df-top 23219  df-bases 23271  df-2ndc 23765

This theorem is used by:  2ndc1stc  23776  2ndcctbss  23781