Theorem conntop 23715

Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)

Assertion

Ref	Expression
conntop	⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Proof of Theorem conntop

Step	Hyp	Ref	Expression
1		eqid 2761	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23711	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 502	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∩ cin 3898  ∅c0 4279  {cpr 4586  ∪ cuni 4867  ‘cfv 6531  Topctop 23191  Clsdccld 23314  Conncconn 23709

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-ext 2733

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-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  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-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-conn 23710

This theorem is used by:  conncompss  23731  txconn  23988  qtopconn  24008  ufildr  24230  connpconn  35969  cvmliftmolem1  36015  cvmliftmolem2  36016  ordtopconn  37197