Theorem 2ndctop 23758

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 23757	. 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 23280	. . . . 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 2862	. . 3 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → 𝐽 ∈ Top)
6	5	rexlimiva 3156	. 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 1570   ∈ wcel 2145  ∃wrex 3087   class class class wbr 5103  ‘cfv 6537  ωcom 7875   ≼ cdom 8964  topGenctg 17601  Topctop 23204  TopBasesctb 23256  2^ndωc2ndc 23749

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-topgen 17607  df-top 23205  df-bases 23257  df-2ndc 23751

This theorem is used by:  2ndc1stc  23762  2ndcctbss  23767